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Functional integral representations for quantum many-particle systems Blois, Cindy Marie 2015

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Functional Integral Representations forQuantum Many-Particle SystemsbyCindy Marie BloisB.Sc. Honours in Physics, The University of Alberta, 2006B.Sc. Honours in Mathematics, The University of Alberta, 2007M.Sc., The University of Toronto, 2008a thesis submitted in partial fulfillmentof the requirements for the degree ofDoctor of Philosophyinthe faculty of graduate and postdoctoralstudies(Mathematics)The University of British Columbia(Vancouver)July 2015c© Cindy Marie Blois, 2015AbstractFormal functional integrals are commonly used as theoretical tools and assources of intuition for predicting phase transitions of many-body systemsin Condensed Matter Physics. In this thesis, we derive rigorous versions ofthese functional integrals for two types of quantum many-particle systems.We begin with a brief review of quantum statistical mechanics in Chapter2 and the formalism of coherent states in Chapter 3, which form the ba-sis for our analysis in Chapters 4 and 5. In Chapter 4, we study a mixedgas of bosons and/or fermions interacting on a finite lattice, with a generalHamiltonian that preserves the total number of particles in each species. Werigorously derive a functional integral representation for the partition func-tion, which employs a large-field cutoff for the boson fields. We then expandthe resulting “action” in powers of the fields and find a recursion relation forthe coefficients. In the case of a two-body interaction (such as the Coulombinteraction), we also find bounds on the coefficients, which give a domain ofanalyticity for the action. This domain is large enough for use of the action inthe functional integral, provided that the large-field cutoffs are taken to grownot too quickly. In Chapter 5, we study a system of electrons and phononsinteracting in a finite lattice, using the Holstein Hamiltonian. Again, werigorously derive a coherent-state functional integral representation for thepartition function of this system and then prove that the “action” in thefunctional integral is an entire-analytic function of the fields. However, sincethe Holstein Hamiltonian does not preserve the total number of bosons, theapproach from Chapter 4 requires some modification. In particular, we re-peatedly use Duhamel expansions in powers of the interaction, rather thansums over particle numbers.iiPrefaceChapters 1, 2 and 3 consist of introductory material to provide backgroundand motivation. No original results are presented in these chapters.All research presented in Chapters 4 and 5 was designed, carried out, andanalyzed by the author.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Quantum Many-Particle Systems . . . . . . . . . . . . . . . . 102.1 Introduction to Quantum Mechanics . . . . . . . . . . . . . . 102.1.1 n-Particle Systems . . . . . . . . . . . . . . . . . . . . 132.2 The Fock Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 202.2.1 Self-Adjoint Operators on Fock Space . . . . . . . . . . 222.2.2 Creation and Annihilation Operators . . . . . . . . . . 252.3 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . 292.3.1 Density Matrices . . . . . . . . . . . . . . . . . . . . . 292.3.2 Grand Canonical Partition Function . . . . . . . . . . 303 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Bosonic Coherent States . . . . . . . . . . . . . . . . . . . . . 323.2 The Grassmann Algebra . . . . . . . . . . . . . . . . . . . . . 35iv3.3 Fermionic Coherent States . . . . . . . . . . . . . . . . . . . . 413.4 Coherent States for Mixed Systems . . . . . . . . . . . . . . . 433.5 Approximate Resolutions of the Identity . . . . . . . . . . . . 463.5.1 For Bosons . . . . . . . . . . . . . . . . . . . . . . . . 463.5.2 For Fermions . . . . . . . . . . . . . . . . . . . . . . . 483.5.3 For Mixed Systems . . . . . . . . . . . . . . . . . . . . 503.6 Trace Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.6.1 For Bosons . . . . . . . . . . . . . . . . . . . . . . . . 533.6.2 For Fermions . . . . . . . . . . . . . . . . . . . . . . . 533.6.3 For Mixed Systems . . . . . . . . . . . . . . . . . . . . 544 Particle-Number-Preserving Interactions . . . . . . . . . . . 574.1 The Physical Setting and Notation . . . . . . . . . . . . . . . 574.1.1 The Hilbert Space . . . . . . . . . . . . . . . . . . . . 574.1.2 Notation for Particle Coordinates and Fields . . . . . . 584.1.3 Creation and Annihilation Operators . . . . . . . . . . 634.1.4 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . 634.2 A Functional Integral Representation for the Partition Function 684.3 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.3.1 Bounds on Fn for 2-Body Interactions . . . . . . . . . 855 Systems of Electrons and Phonons . . . . . . . . . . . . . . . 955.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.2 The Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 985.2.1 Bounds on K . . . . . . . . . . . . . . . . . . . . . . . 1035.3 A Functional Integral Representation for the Partition Function1045.4 The Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142vList of FiguresFigure 1.1 The naïve effective potential, for positive and negativechemical potentials µ. . . . . . . . . . . . . . . . . . . . . 6viList of SymbolsThe table below is serves as a supplemental reference to keep track of thesymbols throughout the thesis. To see each definition or notation in context,please click the corresponding hyperlinked page number below.Symbol Page Meaningσ(x1, . . . , xn) 15 (xσ(1), . . . , xσ(n))(σf)(x1, . . . , xn) 15 f(xσ−1(1), . . . , xσ−1(n))Bn(X) 16 {f ∈ L2(Xn) : (∀σ ∈ Sn)σf = f}Fn(X) 16 {f ∈ L2(Xn) : (∀σ ∈ Sn)σf = (−1)σf}S(n)b (X) 161n!∑σ∈Sn σS(n)f (X) 161n!∑σ∈Sn(−1)σσN0 17 {0, 1, 2, . . . }u⊗s v 18 1√m!(m+n)!n!∑σ∈Sm+n σ(u⊗ v)u ∧ v 18 1√m!(m+n)!n!∑pi∈Sn+m(−1)pipi(u⊗ v)δY 20 cY δy1 ⊗s · · · ⊗s δyn (for bosons)cY 20 (∏x∈X µY (x)!)− 12µY (x) 20 the number of times x appears in YδY 20 δy1 ∧ · · · ∧ δyn (for fermions)Dn(X) 21 L2(Xn)viiD(X) 21⊕∞n=1Dn(X)B(X) 21⊕∞n=1 Bn(X)F(X) 21⊕|X|n=1Fn(X)Sb(X) 22⊕∞n=1 S(n)b (X)Sf (X) 22⊕∞n=1 S(n)f (X)P (n) 22 projection onto the n-particle subspaceD(finite)(X) 23 {f ∈ D(X) : {n : P (n)f 6= 0} is finite}|φ〉 34, 41 e∫Xdy φ(y)a†γ(y)|0〉 (where γ = b or f)H 44 B ⊗ FB 44 B(X1)⊗ · · · ⊗ B(XPb)F 44 F(XPb+1)⊗ · · · ⊗ F(XP )Bn 44 Bn1(X1)⊗ Bn2(X2)⊗ · · · ⊗ BnPb (XPb)B≤n 44 (⊕n1i=0 Bi(X1))⊗ · · · ⊗(⊕nPbi=0 Bi(XPb))m ≤ n 44 (∀p)mp ≤ npH≤n 44 B≤n ⊗FHn 44 Bn ⊗FPn 44 the projection from H onto H≤nP (n) 44 the projection from H onto Hnφ 45 (φ1, . . . , φPb , φPb+1, . . . , φP )|φ〉 45 |φ1〉 ⊗ |φ2〉 ⊗ · · · ⊗ |φP 〉Ir 47∫L2(X) dµr(φ∗, φ)e−‖φ‖2|φ〉〈φ|Ir 51∫dµr(φ∗,φ)e−∫dx φ∗(x)·φ(x)|φ〉〈φ|Xn/Sn 55 Xn11 /Sn1 ×Xn22 /Sn2 × · · · ×XnPbPb /SnPbx ◦ y 59 (~x1 ◦ ~y1, ~x2 ◦ ~y2, . . . , ~xP ◦ ~yP )~xp ◦ ~yp 59 (xp1 , xp2 , . . . , xpnp , yp1 , yp2 , . . . , ypmp )viiiXn 59 Xn12 × · · · ×XnPPX 59⋃n∈NP0Xn|n| 59∑Pp=1 npIx 60 the set of all indices of components of x~x[≤m] 60 (x1, . . . , xm)x[≤m] 61 (~x1[≤m1] , . . . , ~xP[≤mP ])x[>m] 61 (~x1[>m1] , . . . , ~xP[>mP ])α∗(x) 61 α∗1(~x1) · · ·α∗P (~xP )α∗p(~xp) 61 α∗p(xp1)α∗p(xp2) · · ·α∗p(xpnp )∂∂α∗(x) 61∂∂α∗P (~xP )· · · ∂∂α∗1(~x1)∂∂α∗p(~xp)62 ∂∂α∗p(xp,np )∂∂α∗p(xp,np−1)· · · ∂∂α∗p(xp,1)n! 62 n1!n2! · · ·nP !Sn 62 Sn1 × Sn2 × · · · × SnP(−1)σ 62∏Pp=Pb+1(−1)σpS(n) 62 1n!∑σ∈Sn(−1)σσa†p(~x) 63 a†p(x1)a†p(x2) · · · a†p(xn)ap(~x) 63 ap(xn)ap(xn−1) · · · ap(x1)a†(x) 63 a†1(~x1)⊗ · · · ⊗ a†P (~xP )a(x) 63 a1(~x1)⊗ · · · ⊗ aP (~xP )Np 63∫Xp dx a†p(x)ap(x)M 64 the “order” of the interactionkL 68−PbB24A , a lower bound for Kep 78 (0, . . . , 0, 1, 0, . . . , 0) with 1 in pth coordinateλv 90 2P+2(∑p,q‖vp,q‖1,∞) ∑06=m∈NP01|m|P+1ixCv 99 2|Xb||Xf |‖v‖∞κL 103λb−µb2 +(2|Xb||Xf |‖v‖∞)2γ , a lower bound for KΩn(t) 105 {(s1, . . . , sn) ∈ [0, t]n :∑nk=1 sk ≤ t}Cb 107 min{√λb − µb, 1}C2(t) 107 4CbCve14 t(λb−µb) max{t,√t}c2 125 4CvCb e14 (λb−µb)xAcknowledgementsI would like to express my utmost gratitude to my advisor, Joel Feldman,for guiding and supporting me, meticulously reading my work, and encour-aging all of my pursuits. I have never known at once a more kind, patient,hard-working, brilliant and humble person. This project would not have beenpossible without his guidance.I am also grateful to David Brydges and Richard Froese for serving on myadvisory committee, for their advice and careful reading of my dissertation.For other conversations related this work, I thank Martin Lohmann, RolandBauerschmidt, and Manfred Salmhofer.I thank all of my friends and family for their emotional support, especiallySophie Lanthier, Michael Souza, Raimundo Briceño, Trevor Blois, GrandmaBetty Blois, and my parents, Ken and Diane Blois.For their generous financial support, I thank the Vanier Canada GraduateScholarship Program, the Natural Sciences and Engineering Research Councilof Canada, and the Killam Trusts.xiDedicationTo Mom and Dad.xiiChapter 1IntroductionFunctional integrals have become a ubiquitous tool in theoretical and mathe-matical physics, providing new intuition, connections between different prob-lems, and stepping stones for further analysis. This thesis is dedicated to rig-orously constructing such integrals for two classes of quantum many-particlesystems. Before outlining our goals and strategies, we will introduce func-tional integrals through three main examples: the Feynman path integral,the Feynman-Kac formula and the coherent-state formal functional integralfor the partition function of a boson gas.Consider a single particle of mass m allowed to move in space R3 underthe influence of a force corresponding to the potential energy V : R3 → R.In quantum mechanics, all information about the motion of this particle iscontained in its wave function, ψt, which is an element of the Hilbert spaceL2(R3), for each time t ≥ 0. For example, the probability of finding theparticle within a region Λ ⊂ R3 at time t is∫Λ|ψt(x)|2 dx. The time evolutionof the wave function ψt is governed by the Schrödinger equation,i~∂∂tψt = Hψt, (1.1)for all t > 0. The Hamiltonian operator H represents the energy of systemand is defined byH := −~22m∆ + V.Here, ∆ is the Laplacian on R3 and V is the multiplication operator cor-1responding to the potential energy function of the same name. (Note thatthe domain of H surely does not contain all of L2(R3). However, it is notnecessary to precisely define it for introductory purposes.) For convenience,we will let ~ = 1 and H0 := − 12m∆, so that H = H0 + V . The solutions of(1.1) must satisfyψt = e−itHψ0.This formula is difficult to work with in practice, because it involves theunbounded operator H. We would like to simplify it into an integral equationof the form,ψt(x) =∫R3dy I(x, y; t)ψ0(y),where I(x, y; t) is the integral kernel of the operator e−itH . Although we donot have a formula for I(x, y; t) offhand, we can use Fourier transforms [1]to prove that the integral kernel of the free time-evolution operator e−itH0 isI0(x, y; t) =( m2piit)3/2eim2t |x−y|2. (1.2)Unfortunately, becauseH0 and V do not commute, we cannot simply separatethe exponential of their sum into a product of their exponentials. That is,e−it(H0+V ) 6= e−itH0e−itV .However, this formula can be modified to a correct form,e−it(H0+V ) = limn→∞(e−itnH0e−itn V)n,which is an instance of the Trotter product formula [2]. The limit is takenin the strong operator topology. Hence,ψt = e−itHψ0 = limn→∞(e−itnH0e−itn V)nψ0.We can now use (1.2) to replace each e−itnH0 with an integral, so that for anyx0,ψt(x0) = limn→∞(m2pii tn) 32n ∫R3ndx1 · · · dxn eim2(t/n) |x0−x1|2e−itn V (x1) · · ·2· · · eim2(t/n) |xn−1−xn|2e−itn V (xn)ψ0(xn)= limn→∞( mn2piit) 32n∫R3ndx1 · · · dxn eiSn(x0,··· ,xn;t)ψ0(xn), (1.3)whereSn(x0, · · · , xn; t) :=n∑j=1tnm2(|xj−1 − xj|t/n)2− V (xj) .Everything up to this point can be proven rigorously. Now, formally takingthe limit as n → ∞, one could imagine that the sequences {x0, · · · , xn}would become continuous paths X : [0, t] → R3 with X(0) = x0 and thatSn(x0, · · · , xn; t) would become the action over each path,S(X; t) :=∫ t0dτ(m2X ′(τ)2 − V (X(τ))).Assuming that the (complex and divergent) factor(nm2piit) 32n could be absorbedinto the “measure”, which we will call dX, we would findψt(x0) =∫Ωx0dX eiS(X;t)ψ0(X(t)). (1.4)“Path integral” formulas such as this were first invented in the context ofquantum mechanics by Richard Feynman in his 1942 PhD thesis. Physicistsnow routinely use this formula and others like it as a symbolic tool in orderto gain intuition and do formal calculations by expanding about classicalpaths, which pleasantly arise as stationary points for the quantum actionS. However, unfortunately, as n → ∞,(mn2piit) 32n dx1 · · · dxn does not con-verge to a measure, both because there is no infinite-dimensional analogueof Lebesgue measure and because the factor in front,(mn2piit) 32n blows up asn → ∞. Furthermore, without restrictions on the paths X, the integrandfluctuates wildly, if it is defined at all. Hence, “dX” in (1.4) is not actuallya measure and the integral is not well-defined. There have been some effortsto create a new definition that incorporates (1.4) and rigorously allows forexpansion about classical paths, for example in [3].If we consider the above analysis in “imaginary time”, i.e. for the operator3e−tH rather than e−itH , the same work carries through to obtain the followinganalogue of (1.3),(e−tHψ0)(x0) = limn→∞(mn2pit) 32n∫R3ndx1 · · · dxn e−S˜n(x0,··· ,xn;t)ψ0(xn), (1.5)whereS˜n(x0, · · · , xn; t) =n∑j=1tnm2(|xj−1 − xj|t/n)2+ V (xj) .The critical insight here is that the integrand now contains a product ofGaussians (rather than the wildly oscillatory factors of eim2t |x−y|2above) dueto the contribution of the imaginary-time free propagators. These can becombined with the Lebesgue measure on R3n to form the 3n-dimensionalGaussian measure,dµn(x1, . . . , xn) =n∏j=1[(mn2pit) 32e−m2|xj−1−xj |2t/n dxj].In the limit n→∞, this measure converges to a probability measure knownas theWiener measure µx0 on the space Ω of continuous paths X : [0, t]→ R3[1]. (This measure had already been invented by Wiener in the early 1920s todescribe Brownian motion [4].) Assuming that V is sufficiently “nice” (e.g.in L2(R3) + L∞(R3) [1]), (1.5) converges to(e−tHψ0)(x0) =∫Ωdµx0(ω) e−∫ t0dτ V ((ω(τ))ψ0(ω(t)).This functional integral is known as the Feynman-Kac formula. It has appli-cations in many different fields, including quantum physics, PDEs, probabil-ity, and finance, thereby providing connections between these fields [5]. Forexample, the Feynman-Kac formula is a solution to the PDE,∂ψt∂t−12m∆ψt + V ψt = 0.This is a heat equation, where V represents a source of external cooling. Onthe other hand, the Feynman-Kac formula may be viewed as an expectationvalue with respect to the Wiener measure, which describes Brownian motion.4This link makes it possible to use properties of the PDE to deduce propertiesof Brownian motion, and vice versa [8]. Particularly relevant to this thesis(although we do not use them here) are applications of Feynman-Kac for-mulae to many-particle systems. For example, Ginibre [6] and Brydges andFederbush [7] constructed Feynman-Kac formulae for N particles and thensummed over all N to obtain the partition function and correlation functionsfor the full Fock space, in the case where the chemical potential is negativeand sufficiently large in magnitude.There is another approach to constructing functional integrals for quan-tum many-particle systems, wherein fields are introduced through “coherentstates”. Coherent-state functional integrals appear to be the most promis-ing tool for rigorously constructing the many-particle models, because they(like the Feynman path integrals described above) naturally contain an ac-tion which provides intuition about the low-energy behaviour of the systemand suggests phase transitions. To see this roughly and quickly, consider asystem of interacting bosons in R3. Through formal manipulations [9], thecoherent-state approach yields a formal functional integral for the partitionfunction of the following form [10]:Z =∫· · ·∫ ∏x∈R30≤τ≤βdφ∗τ (x)dφτ (x)2piieA(φ∗,φ), (1.6)where the domain is over all fields φτ : R3 → C, as functions of the (imag-inary) time-like variable τ ∈ [0, β], with the periodic boundary condition,φ0 = φβ. For each τ ∈ [0, β] and x ∈ R3,∫ dφ∗τ (x)dφτ (x)2irepresents an integral over C ∼= R2 with Lebesgue measure. That is, forz = u+ iv, dz∗dz2i = du dv.As in the Feynman path integral (1.4), the exponent A(φ∗, φ) in Equation(1.6) is called the “action” and it depends on all φτ (x), φ∗τ (x). In particular,A(α∗, φ) :=∫ β0dτ∫R3dx α∗τ (x)∂∂τφτ (x)−∫ β0dτ K(α∗τ , φτ ),5µ < 0µ > 0Re(Φ)Figure 1.1: The naïve effective potential, for positive and negativechemical potentials µ.whereK(α∗, φ) =∫R3dx∫R3dy α∗(x)h(x, y)φ(y)− µ∫R3dx α∗(x)φ(x) (1.7)+∫R3dx∫R3dy α∗(x)α∗(y)v(x, y)φ(x)φ(y).Here, h is a kernel for the single-particle kinetic energy operator, µ is thechemical potential, and v(x, y) represents the potential energy due to a par-ticle at position x interacting with a particle at position y.The formal functional integral (1.6) provides intuition about phase transi-tions of the system. To see this briefly, assume for simplicity that the fieldφ is a constant, that is φτ (x) = Φ ∈ C for all τ and x. Then the actionsimplifies to a function of the form,A(Φ∗,Φ) = −∫ β0dτ∫R3dx(v0|Φ|4 − µ|Φ|2),where v0 =∫R3 dy v(x, y). So A(Φ∗,Φ) is minus the volume of “space-time”(which is infinite, but we’ll ignore that) times the “naïve effective potential”,v0|Φ|4 − µ|Φ|2. When we look for its stationary points, we see two cases:when µ ≤ 0, the only minimum occurs at Φ = 0; otherwise, if µ > 0, thereis a degenerate minimum on the circle |Φ| =√µ2v0. This is depicted in Fig-ure 1.1. Hence at low temperatures, we expect the fields to be near zero6when µ < 0 and nonzero when µ > 0. This signals a phase transition.Unfortunately, the formal functional integral in (1.6) is not well-defined.Since the domain is (uncountably) infinite-dimensional, there is no clear ex-tension of the Lebesgue measure to it. In the spirit of the Wiener measure,one could hope to absorb part of the exponential, eA(φ∗,φ), to construct aproper infinite-dimensional Gaussian measure, however, the purely imaginaryterm∫ β0 dτ∫R3 dx α∗τ (x)∂∂τ φτ (x) oscillates so wildly that this is not possible,as shown in Appendix A of [10].In this thesis, our goal is to rigorously construct functional integrals of theform in Equation (1.6). Our main strategy is to:1. approximate the physical space R3 by a finite set X,2. approximate the time-like interval [0, β) by a finite set Tq = {, 2, . . . , q= β},3. use a large-field cutoff in the integral, so that rather than integratingover the whole complex plane for each τ and x, we only integrate overa ball of finite radius, |φτ (x)| < R.To recover (1.6), we would like to take the limits as the size of X grows toinfinity, as q → ∞, and as R → ∞. In this work, we take the limit as bothq and R grow to infinity at the same time, considering R as a function of q.However, we leave the limit |X| → ∞ for future work.We consider two classes of many-body systems, both of which consist ofmixed gases of bosons and fermions. First, we study a system of any finitenumber of particle species (bosons and/or fermions), interacting on a finitelattice with a general Hamiltonian that preserves the total number of parti-cles in each species. This is a generalization of the work for a purely bosonicgas in [10]. Gases of this type have been predicted and observed to haveremarkably rich phase diagrams, including such features as superfluidity, su-persolidity, and phase separation [11–13]. Next, we consider a boson-fermiongas with an interaction that does not preserve the number of bosons. TheHamiltonian is called the “Holstein Hamiltonian” [14] and has been usedextensively to model systems of electrons and phonons [15]. In these sys-tems, the phenomena of “polarons” (quasiparticles of electrons surrounded7by a cloud of phonons) and phonon-mediated superconductivity have gar-nered much attention and research. Furthermore, this type of Hamiltonianarises when a Hubbard-Stratonovich transformation [16] is performed on thepartition function for a purely fermionic gas with a two-body interaction.We begin in Chapter 2 with a brief overview of background material, includ-ing non-relativistic quantum mechanics, Fock spaces, and quantum statisti-cal mechanics. In Chapter 3, we define the coherent states and develop atheoretical toolkit, which includes a resolution of the identity and a traceformula. We follow the approaches of [9] and [10] for the coherent states ofsingle species of bosons and fermions, and then extend to gases with multiplespecies of bosons and/or fermions. Our main results lie in Chapters 4 and5. Theorem 4.2.1 gives a functional integral representation for the partitionfunction of any quantum system with a fixed and finite number of parti-cle species (bosons, fermions, or both) interacting on a finite lattice, witha Hamiltonian that preserves the total number of particles for each species.This is of the form,Tr e−βK = limq→∞∫ ∏τ∈Tq[dµR(q)(φ∗τ ,φτ )e−∫dy φ∗τ (y)·φτ (y)] ∏τ∈Tq〈φτ−|e−K |φτ 〉,(1.8)where β is the inverse temperature, K = H − µ ·N, H is the Hamiltonian,µ is the vector of chemical potentials, N is the vector of particle numberoperators, and R(q) represents a large-boson-field cutoff that grows suffi-ciently quickly as q →∞. Each |φτ 〉 is a coherent state (which is defined inChapter 3). In Lemma 4.3.3, we expand the “action”, i.e. the logarithm of〈α|e−K |φ〉, in powers of the fields,log〈α|e−K |φ〉 =∑n∈NP0∫Xndx∫Xndy α∗(x)Fn(,x,y)φ(y),and derive a recursion relation for the coefficients Fn(,x,y) in Theorem4.3.3. For two-body interactions (such as the Coulomb interaction), we haveshown in Theorem 4.3.5 that the coefficients Fn(,x,y) decay as n grows. InProposition 4.3.6, these bounds are then used to find a domain of analyticityfor the action. This domain is large enough such that an exponential of theaction may be substituted for each inner product in (1.8) above, providedthat the large-field cutoffs are taken to be not too large.8In Chapter 5, we study the electron-phonon model. Because the Hamilto-nian does not preserve the number of bosons, we require extra control overexpansions in boson number. For this, we generally use an expansion of e−Kin powers of the interaction V . This is called a Duhamel expansion and isshown in Theorem 5.3.1. In Theorem 5.3.8, we find a functional integralrepresentation for the partition of this system, which is also of the form in(1.8) above. Finally, we prove in Section 5.4 that the action exists as anentire-analytic function of the fields.9Chapter 2Quantum Many-ParticleSystemsThis chapter gives a rapid overview of the quantum theory that underpinsthis thesis. Readers who have already been introduced to annihilation andcreation operators may skip to Section 2.3, and those who are already familiarwith the grand canonical ensemble may skip straight to Chapter 3.2.1 Introduction to Quantum MechanicsEvery quantum system is associated with a Hilbert space, H. Each ray ofthis Hilbert space represents a state of the system. A ray is defined to be anequivalence class of the relation ‘∼’, where, for any ψ, φ ∈ H,ψ ∼ φ :⇐⇒ (∃λ ∈ C\{0}) ψ = λφ.The set of all rays is the projective Hilbert space,PH := H/ ∼ .On the other hand, every pure state of the quantum system may be repre-sented by an element of PH. For convenience, we will often denote elementsof the projective space (and hence, pure states) by a single representativeelement ψ ∈ H with norm ‖ψ‖ = 1. (The meaning of the word “pure” inthis context will be explained in section 2.3.1.)10Physically, an observable is any quantity of the system which may be mea-sured. Examples include position, momentum, energy, and angular momen-tum. Mathematically, each observable is represented by a self-adjoint opera-tor on H, or a dense subspace thereof. Measurement of the observable yieldsa real number that lies in the spectrum of this operator.When the system is in state ψ ∈ H (with ‖ψ‖ = 1), the expectation value ofthe observable corresponding to the self-adjoint operator A is〈ψ,Aψ〉.By the spectral theorem [2], each self-adjoint operator A is associated witha projection-valued measure,PS = χS(A),defined for Borel sets S ⊂ R. Here, χS is the characteristic function,χS(x) :=1 if x ∈ S,0 otherwise.The probability of measuring the corresponding observable to lie within aBorel set S ⊂ R while the system is in state ψ is〈ψ, χS(A)ψ〉 = 〈ψ, PS ψ〉 =∫Sd〈ψ, Pλ ψ〉.A self-adjoint operator of particular interest is the one corresponding to thetotal energy of the system, known as the Hamiltonian, because it generatestime evolution. That is, the state ψ of a quantum system evolves in time taccording to the Schrödinger equation,i~dψdt= Hψ,where ~ is a constant with dimensions of [energy] × [time] and H is theHamiltonian. This equation can be derived from the principles of causality,superposition, and correspondence with classical mechanics in the classicallimit [17].11The vocabulary thus far can be summarized as follows:quantum system ←→ Hilbert space H,pure state ←→ element of PH,observable ←→ self-adjoint operator on H.Since we are working from the mathematical side in this thesis, we mayconsider the words on the left as vocabulary for the mathematical objectson the right, even though they also correspond with tangible entities of thephysical world. Without attempting the philosophical task of clarifying thiscorrespondence, we refer to an example to help the reader gain intuition.Example 2.1.1 (A free particle in 1 dimension). Consider a quantum systemconsisting of a single particle allowed to move freely within some interval ofa 1-dimensional space, Ω ⊂ R. The Hilbert space for this system is L2(Ω),the set of all square-integrable functions on Ω with Lebesgue measure. Theoperator corresponding to the position of the particle is multiplication by theidentity function,(xˆψ) (x) := xψ(x),for any x ∈ Ω and ψ ∈ L2(Ω) such that the function on the right side is alsoan element of L2(Ω). For example, if Ω is compact, xˆ is well-defined as abounded operator on all of L2(Ω). If the system is in state ψ ∈ L2(Ω) with‖ψ‖ = 1, then the expected value of the position is〈ψ, xˆψ〉 =∫Ωx|ψ(x)|2 dx.This means that if we were to prepare many independent isolated copies ofthis quantum system, all in state ψ, and we measured the position of theparticle in each, we would expect the mean value of all such measurementsto be∫Ωx|ψ(x)|2 dx. The probability of finding the particle within the(Borel) set Γ ⊂ Ω is〈ψ, χΓ(xˆ)ψ〉 =∫Ωψ∗(x)χΓ(x)ψ(x) dx =∫Γ|ψ(x)|2 dx.In this sense, the state ψ gives rise to a probability distribution |ψ(x)|2 forthe position of the particle.12If Ω = R, the Hamiltonian isH = −~22md2dx2,defined on the domain C2(Ω), which is dense in L2(Ω). If the derivativesare taken to be weak derivatives, the domain may also be extended to theSobolev space H2(Ω) [17].Suppose that the system begins in state ψ0 ∈ L2(R) and evolves in time.The state at time t is denoted by ψt ∈ L2(R). For any x ∈ R, we use thenotation ψ(x, t) := ψ(t)(x). The time evolution is given by the Schrödingerequation,i~∂ψ∂t(x, t) = −~22m∂2ψ∂x2(x, t), (2.1)with the initial condition,ψ(x, 0) = ψ0(x)for all x ∈ R. Since Equation (2.1) is a type of wave equation, it is commonto call the solution ψ a wave function.2.1.1 n-Particle SystemsFor a quantum system consisting of a single particle in a region Ω of space, thecorresponding Hilbert space is L2(Ω). What is the Hilbert space for two ormore particles in Ω? More generally, if two distinguishable quantum systemswith Hilbert spaces H1 and H2 are combined and considered altogether as asingle system, what is the Hilbert space of the composite system? The Hilbertspace H of the composite system is the tensor product of its constituentspaces,H = H1 ⊗H2.This is a tensor product of Hilbert spaces, which is similar to a vector-space tensor product, but completed under the topology induced by theinner product,〈φ1 ⊗ φ2, ψ1 ⊗ ψ2〉 = 〈φ1, ψ1〉〈φ2, ψ2〉,for all φ1, ψ1 ∈ H1 and φ2, ψ2 ∈ H2. (A full definition of the Hilbert spacetensor product may be found in standard textbooks such as [2].)13Successively combining quantum systems with Hilbert spaces H1, H2, . . . ,Hn, the resulting Hilbert space for the n-component composite system isH1 ⊗H2 ⊗ · · · ⊗ Hn.Back to our original question of this section, consider a quantum systemconsisting of n distinguishable particles in a region Ω. The correspondingHilbert space is thenDn(Ω) := L2(Ω)⊗ L2(Ω)⊗ · · · ⊗ L2(Ω)︸ ︷︷ ︸n times∼= L2(Ωn).In this thesis, we are also interested in quantum systems of indistinguishableparticles. Aiming to define the corresponding Hilbert spaces, let’s intuitivelystart by considering a state ψ ∈ L2(Ω2) for a quantum system of two dis-tinguishable particles in Ω and then ask what we should demand of ψ if theparticles were to become indistinguishable.First, indistinguishability should imply that no measurement is affected bythe permutation of particle coordinates. For example, let ψs be the functionobtained by interchanging the two particle coordinates,ψs(x1, x2) = ψ(x2, x1).We demand that the expectation value of any observable A be unaffected bythis interchange,〈ψ,Aψ〉 = 〈ψs, Aψs〉.Since this is true for any observable A, it follows that|ψ(x1, x2)|2 = |ψs(x1, x2)|2for all x1, x2 ∈ Ω. Hence, the two functions agree up to a phase shift,ψ(x2, x1) = eiθ(x1,x2)ψ(x1, x2).for some θ(x1, x2) ∈ R. In three space-dimensions, θ is a constant, equal to0 or pi. That is,ψs = ±ψ,14which means that the wave functions are either symmetric or antisymmet-ric. Particles with symmetric wave functions are said to satisfy Bose-Einsteinstatistics and are called bosons. Those with antisymmetric wave functions aresaid to satisfy Fermi-Dirac statistics and are called fermions. Furthermore,it can be proven that there is a direct link between the spin of a particleand its statistics. This remarkable result, called the Spin-Statistics Theo-rem, verifies the experimental observation that particles of integer spin arebosons, whereas particles with half-integer spin are fermions [18].We can extend this intuition to any number n of indistinguishable particlesin three dimensions by demanding that their wave functions ψ ∈ L2(R3n)satisfyψ(xσ(1), . . . , xσ(n)) = (±1)σψ(x1, . . . , xn),for all (x1, . . . , xn) ∈ R3n and every permutation σ ∈ Sn. Again, the ± signon the right side may be read as “+” for bosons and “−” for fermions, where(+1)σ ≡ 1 and (−1)σ is the sign of the permutation σ.We will use the following notation for the action of permutations on n-particlecoordinates and wave functions.Definition 1. Let X be any set.(i) For any n ∈ N ∪ {0} and σ ∈ Sn, letσ : Xn → Xn,(x1, . . . , xn) 7→ (xσ(1), . . . , xσ(n)). (2.2)(ii) For any n ∈ N ∪ {0} and σ ∈ Sn, letσ : L2(Xn)→ L2(Xn),f 7→ f ◦ σ−1.That is,(σf)(x1, . . . , xn) = f(xσ−1(1), . . . , xσ−1(n)), (2.3)for any (x1, . . . , xn) ∈ Xn.We are now prepared to define the Hilbert spaces for n indistinguishableparticles.15Definition 2. Let (X,µ) be a measure space.(i) For any n ∈ N, the Hilbert space for n identical bosons in X isBn(X) := {f ∈ L2(Xn) : (∀σ ∈ Sn)σf = f}, (2.4)the set of fixed points for the action of Sn on L2(Xn), otherwise knownas the symmetric functions from Xn to C.(ii) For any n ∈ N, the Hilbert space for n identical fermions in X isFn(X) := {f ∈ L2(Xn) : (∀σ ∈ Sn)σf = (−1)σf}, (2.5)the set of antisymmetric functions from Xn to C.(iii) The vacuum is the normalized state with zero particles,|0〉 := 1 ∈ C.We also define the zero-boson and zero-fermion spaces to beB0(X) = F0(X) := C.For all n, Bn(X) and Fn(X) are subspaces of L2(Xn), closed under the innerproduct〈f, g〉 :=∫Xnf ∗g dµ,where f ∗ is the complex conjugate of f . The orthogonal projection ontoBn(X) is the symmetrization operator,S(n)b (X) :=1n!∑σ∈Snσ. (2.6)Likewise, the anti-symmetrization operator,S(n)f (X) :=1n!∑σ∈Sn(−1)σσ (2.7)is the orthogonal projection onto Fn(X).16Just as we combine Hilbert spaces corresponding to distinguishable quantumsystems using a tensor product, indistinguishable systems can be combinedby taking a tensor product of their Hilbert spaces and then symmetrizing oranti-symmetrizing. That is, for any m,n ∈ N0 (where N0 := {0, 1, 2, . . . },the set of whole numbers),Bn+m(X) = S(n+m)b (X) Bn(X)⊗ Bm(X)andFn+m(X) = S(n+m)f (X) Fn(X)⊗Fm(X).We also define corresponding symmetric and antisymmetric tensor productsfor elements of these spaces. Suppose that u ∈ L2(Xn) and v ∈ L2(Xm) forsome n,m ∈ N0. Their tensor product u⊗ v in L2(Xn+m) is given byu⊗ v(x1, . . . , xn, xn+1, . . . , xn+m) = u(x1, . . . , xn)v(xn+1, . . . , xn+m),for all (x1, . . . , xn, xn+1, . . . , xn+m) ∈ Xn+m. Aiming to define symmetric andanti-symmetric versions of this tensor product that lie in the bosonic andfermionic subspaces respectively, our first thought might be to simply takethe tensor product above and then symmetrize or antisymmetrize. Clearly,S(n+m)b (X)(u⊗ v) ∈ Bn+m(X) and S(n+m)f (u⊗ v) ∈ Fn+m(X).We could simply define these as the symmetric and antisymmetric tensorproducts of elements u and v (and some authors do); however, in order toensure associativity of these products, we will include a combinatorial factor(depending on m and n) in the definitions.Definition 3. For any m,n ∈ N, let u ∈ L2(Xn) and v ∈ L2(Xm).(i) The tensor product of u and v, u⊗ v ∈ L2(Xm+n), is defined byu⊗ v(x1, . . . , xn, xn+1, . . . , xn+m) := u(x1, . . . , xn)v(xn+1, . . . , xn+m),(2.8)for any (x1, . . . , xn, xn+1, . . . , xn+m) ∈ Xn+m.17(ii) The symmetric tensor product of u and v isu⊗s v :=√(m+ n)!m!n!S(m+n)b (X)(u⊗ v)=1√m!(m+ n)!n!∑σ∈Sm+nσ(u⊗ v).(2.9)(iii) The antisymmetric tensor product or wedge product of u and v isu ∧ v :=√(m+ n)!m!n!S(m+n)f (u⊗ v)=1√m!(m+ n)!n!∑pi∈Sn+m(−1)pipi(u⊗ v).(2.10)It is easy to verify that these products are indeed associative. Further-more, the annihilation operators, which we will define in Section 2.2.2, satisfyLeibniz-like product rules with respect to these products. The symmetric ten-sor product is clearly commutative by definition, whereas the wedge productsatisfiesu ∧ v = (−1)nmv ∧ u.for all u ∈ Fn(X) and v ∈ Fm(X).Example 1. If a single particle in state u is combined with with n particles instate v, where all of these particles (including the single) are identical, whatis the resulting (n + 1)-particle state? In the case of bosons, if u ∈ B1(X)and v ∈ Bn(X) for some n ∈ N,u⊗s v(x1, . . . , xn+1) =1√n+ 1n+1∑j=1u(xj)v(x1, . . . , xj−1, xj+1, . . . , xn+1).Similarly, in the case of fermions, if u ∈ F1(X) and v ∈ Fn(X) for somen ∈ N,u ∧ v(x1, . . . , xn+1) =1√n+ 1n+1∑j=1(−1)j−1u(xj)v(x1, . . . , xj−1, xj+1, . . . , xn+1).18Example 2. If n identical particles in states u1, . . . , un ∈ B1(X) = F1(X) =L2(X) are combined, the resulting n-particle state will be as follows: forbosons,u1 ⊗s · · · ⊗s un(x1, . . . , xn) =1√n!∑σ∈Snu1(xσ−1(1)) · · ·un(xσ−1(n)),and for fermions,u1 ∧ · · · ∧ un(x1, . . . , xn) =1√n!∑σ∈Sn(−1)σu1(xσ−1(1)) · · ·un(xσ−1(n)).These results follow from Example 1 by induction.Given an orthonormal basis for the single-particle space L2(X), we can com-bine its elements using the symmetric and anti-symmetric tensor products asin Example 2 in order to construct orthonormal bases for Bn(X) and Fn(X)respectively.Proposition 2.1.2. Let X be a finite set. Suppose that {ϕi : i = 1, . . . , |X|}is an orthonormal basis for the single-particle Hilbert space, L2(X).(i) The set{1√α!ϕ⊗αs : α ∈ N|X|0 , |α| = n}is an orthonormal basis for Bn(X). For the “multi-index” α, we use thefollowing notation:α! := α1!α2! . . . α|X|!,|α| :=|X|∑j=1αj,andϕ⊗αs := ϕ⊗α1s1 ⊗s ϕ⊗α2s2 ⊗s · · · ⊗s ϕ⊗α|X|s|X| .(ii) The set{ϕi1 ∧ ϕi2 ∧ · · · ∧ ϕin : {ij}nj=1 ⊂ {1, . . . , |X|} with i1 < i2 < · · · < in}is an orthonormal basis for Fn(X).19One particularly useful orthonormal basis for L2(X) is the set of Kroneckerdelta functions. For any y ∈ X, the Kronecker delta function δy is given byδy(x) =1 if x = y,0 otherwise,for all x ∈ X. Applying Proposition 2.1.2, we see that{δY : Y ∈ Xn/Sn}is an orthonormal basis for Bn(X), where if (y1, . . . , yn) is any representativefor Y ∈ Xn/Sn, thenδY := cY δy1 ⊗s · · · ⊗s δyn , (2.11)withcY :=1√ ∏x∈XµY (x)!, (2.12)and µY (x) := |{j : x = yj}|, the number of times x appears in Y . Similarly,if we introduce a total ordering (arbitrarily) on X,{δy1 ∧ δy2 ∧ · · · ∧ δyn : y1, . . . , yn ∈ X, yi < yj for all i < j} (2.13)is a basis for Fn(X). If Y = {y1, . . . , yn} ⊂ Xn with ordered elementsy1 < · · · < yn, thenδY := δy1 ∧ · · · ∧ δyn . (2.14)Hence, the basis in Equation (2.13) can also be written as {δY : Y ⊂ X}.By counting the elements of these bases, we can see that the dimension ofBn(X) is(n+|X|−1n)and the dimension of Fn(X) is, when n ≤ |X|,(|X|n).2.2 The Fock SpacesWe are interested in quantum systems where particles may be created anddestroyed, or allowed to move in and out of the system. In this section, we20define the corresponding Hilbert spaces, which must allow for arbitrary andvarying particle numbers.Throughout this section, let X be a finite set with the counting measure.That is, for any function f : X → C,∫Xf(x) dx :=∑x∈Xf(x).Recall that the Hilbert space for a fixed number, n, of distinguishable parti-cles in the region X isDn(X) = L2(X)⊗ · · · ⊗ L2(X)︸ ︷︷ ︸n times= L2(X)⊗n ∼= L2(Xn) ∼= C|X|n. (2.15)Taking the direct sum of all n-particle Hilbert spaces, we obtainD(X) :=∞⊕n=0Dn(X), (2.16)which contains states with arbitrary particle numbers and superpositionsthereof. This Hilbert space is called the Fock space for distinguishable parti-cles.Similarly, the Fock space for indistinguishable bosons is defined to beB(X) :=∞⊕n=1Bn(X), (2.17)and for indistinguishable fermions,F(X) :=∞⊕n=1Fn(X). (2.18)However, since X is finite, there are no nontrivial antisymmetric functionson X with |X|+ 1 arguments or more. Hence, Fn(X) = {0} for all n > |X|.It follows that F(X) is finite-dimensional andF(X) ∼=|X|⊕n=1Fn(X). (2.19)21Each of B(X) and F(X) are closed subspaces of D(X); the orthogonal projec-tions onto these spaces are the direct sums of the n-particle symmetrizationand anti-symmetrization operators respectively. That is, the projection fromD(X) to B(X) is the operatorSb(X) :=∞⊕n=1S(n)b (X), (2.20)where the right-hand side means that the restriction of Sb(X) to each Dn(X)is S(n)b (X). The projection from D(X) to F(X) is the operatorSf (X) :=∞⊕n=1S(n)f (X). (2.21)(Again, the latter sum can be truncated at n = |X|.)We will denote by P (n) the projection from the Fock space D(X) to then-particle subspace, Dn(X). Note that P (n) preserves symmetry and anti-symmetry and therefore can also be seen as the projection from B(X) toBn(X) and from F(X) to Fn(X).Recall that self-adjoint operators in the Hilbert space correspond to “ob-servables”: quantities of the system that may be measured, such as energy,position, and momentum. We will discuss such operators on Fock spaces, aswell as the creation and annihilation operators, which are not self-adjoint,but will provide for a useful representation of the self-adjoint operators ofinterest.2.2.1 Self-Adjoint Operators on Fock SpaceConsider a single particle in the space X (which we are still assuming to bea finite set). Let h be a self-adjoint operator on L2(X) that represents thekinetic energy of the particle. A typical example ish = −12m∆,where m is the mass the particle and ∆ is the discrete Laplacian on X. Forexample, if X is a one-dimensional lattice with spacing δ between lattice22sites, then for any f : X → C,hf(x) = −12m∆f(x) = −12mf(x+ δ)− 2f(x) + f(x− δ)δ2.For n distinguishable particles in X, the total kinetic energy is intuitivelythe sum of the kinetic energies for each particle. This is defined byHn ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψn :=(hψ1)⊗ ψ2 ⊗ · · · ⊗ ψn + ψ1 ⊗ (hψ2)⊗ · · · ⊗ ψn+· · ·+ ψ1 ⊗ ψ2 ⊗ · · · ⊗ (hψn)=n∑j=1ψ1 ⊗ · · · ⊗ (hψj)⊗ · · · ⊗ ψn,for any state of the form ψ1 ⊗ ψ2 ⊗ · · · ⊗ ψn where each ψj ∈ L2(X). Wecan extend by linearity to all of Dn(X). The extension of h to the full Fockspace,dΓ(h) :=∞⊕n=1Hn, (2.22)is known as the second quantization of h. This operator is essentially self-adjoint on the domainD(finite)(X) := {f ∈ D(X) : {n : P (n)f 6= 0} is finite} (2.23)and self-adjoint on the domain,D(dΓ(h)) :={ψ ∈ D(X) :∞∑n=1∥∥∥HnP(n)ψ∥∥∥2<∞}.Example 3. The second-quantization of the identity operator,N := dΓ(1) (2.24)is known as the particle-number operator. For example, if ψ ∈ Dn(X), thenNψ = nψ.In order to account for n-body interactions, we will also need to extendthe corresponding operators on Dn(X), where n ≥ 2, to the Fock space.Most interactions in nature are accurately modelled by two-body potentials.23Suppose that the operator V on L2(X2) represents the energy correspondingto a two-body interaction. We may extend V to the n-particle space (wheren ≥ 2) by summing over particle pairs. For example, suppose that V isa multiplication operator by the function v : X2 → C, such that for allφ ∈ L2(X2),V φ(x1, x2) = v(x1, x2)φ(x1, x2)A physically important case is the Coulomb interaction, in which v has theformv(x1, x2) =kq1q2|x1 − x2|,where q1 and q2 are the charges of the two particles and |x1 − x2| is thedistance between them.For each pair i, j with 1 ≤ i < j ≤ n, letV (ij)ψ(x1, . . . , xn) := v(xi, xj)ψ(x1, . . . , xn),for all ψ ∈ L2(Xn). Then ifVn :=∑1≤i<j≤nV (ij),for all n ≥ 2, the second quantization of V isdΓ(V ) :=∞⊕n=2Vn. (2.25)Again, this operator is essentially self-adjoint on the domain D(finite)(X) andself-adjoint on the domain,D(dΓ(V )) :={ψ ∈ D(X) :∞∑n=2∥∥∥VnP(n)ψ∥∥∥2<∞}.Unless otherwise stated, if A is any operator from Un(X) to Un(X) for alln ∈ N0, where U = D,B, or F , we will assume the domain of A to be the setD(A) ={ψ ∈ U(X) :∞∑n=0∥∥∥AP (n)ψ∥∥∥2<∞}. (2.26)242.2.2 Creation and Annihilation OperatorsLet ψ ∈ Dn(X) be a n-distinguishable-particle state. Roughly speaking, thecreation operator at x ∈ X, a†(x), adds a new particle at the point x. Moreprecisely,a†(x)ψ :=√n+ 1 δx ⊗ ψ ∈ Dn+1(X).On the other hand, the annihilation operator at x ∈ X, a(x), destroys aparticle at the point x. That is, a(x)ψ ∈ Dn−1(X) is defined by[a(x)ψ](x1, . . . , xn−1) =√n ψ(x, x1, · · · , xn−1).Each of these operators can be linearly extended to D(finite)(X).The creation and annihilation operators for bosons and fermions may beobtained from their distinguishable counterparts by symmetrization and anti-symmetrization respectively.Definition 4. (a) The boson creation and annihilation operators are definedbya†b(x) := Sb(X)a†(x)Sb(X) and ab(x) := Sb(X)a(x)Sb(X),respectively.(b) The fermion creation and annihilation operators are defined bya†f (x) := Sf (X)a†(x)Sf (X) and af (x) := Sf (X)a(x)Sf (X),respectively.Since Sf (X) projects D(X) onto a finite-dimensional subspace F(X), af (x)and a(†)f (x) are bounded operators defined on all of D(X). The boson creationand annihilation operators are defined on D(finite)(X) and may be extendedto the following respective domains: for annihilation operators,D(ab(x)) :={u ∈ D(X) :∞∑n=1‖ab(x)P (n)u‖2Bn−1(X) <∞},25and for creation operators,D(a†b(x)) :={u ∈ D(X) :∞∑n=0‖a†b(x)P(n)u‖2Bn+1(X) <∞}.Proposition 2.2.1. The creation operators satisfya†b(x)u = δx ⊗s (Sb(X)u) and a†f (x)u = δx ∧ (Sf (X)u),for all u ∈ D(finite)(X) and x ∈ X.Proposition 2.2.2. The creation operators are the adjoints of their respec-tive annihilation operators. That is, for all x ∈ X, a†f (x) is the adjoint ofaf (x) and a†b(x) is the adjoint of ab(x), provided that the domains of theboson creation annihilation operators are taken to be D(a†b(x)) and D(ab(x))respectively, as defined above.The creation and annihilation operators often provide a convenient and intu-itive way of expressing other many-body operators on the Fock space. Thesecond-quantization of the two-particle operator V , dΓ(V ) from Equation(2.25), may be expressed asdΓ(V ) =∫X2dx dy a†(x)a†(y)v(x, y)a(x)a(y). (2.27)Similarly, the second quantization of the single-particle operator h : L2(X)→L2(X), dΓ(h) from Equation (2.22), satisfiesdΓ(h) =∫X2dx dy a†(x)h˜(x, y)a(y), (2.28)where h˜ is the integral kernel of h. In particular, for the particle-numberoperator, Equation (2.28) reads,N = dΓ(1) =∫Xdx a†(x)a(x).For each x, letn(x) := a†(x)a(x).This is called the particle-density operator and represents the number of par-ticles present at the point x ∈ X.26Similarly, the boson-density operator at x ∈ X isnb(x) := Sb(X)n(x)Sb(X) = a†b(x)ab(x)and the fermion-density operator at x ∈ X isnf (x) := Sf (X)n(x)Sf (X) = a†f (x)af (x).Intuitively, the total-particle-number operator for distinguishable particlessatisfies N =∑x∈X n(x). Similarly, for bosons and fermions, we letNb := Sb(X)NSb(X) =∑x∈Xnb(x)andNf := Sf (X)NSf (X) =∑x∈Xnf (x).Proposition 2.2.3 (Canonical Commutation and Anti-Commutation Rela-tions). The boson creation and annihilation operators satisfy the commuta-tion relations,[ab(x), a†b(y)]− = δx(y),[ab(x), ab(y)]− = 0,[a†b(x), a†b(y)]− = 0,where [A,B]− represents the commutator,[A,B]− = AB −BA,for all operators A and B for which the right side is well-defined.The fermion creation and annihilation operators satisfy the anti-commutationrelations,[af (x), a†f (y)]+ = δx(y),[af (x), af (y)]+ = 0,[a†f (x), a†f (y)]+ = 0.27The anti-commutator [A,B]+ is given by[A,B]+ = AB +BA,for all operators A and B for which the right side is well-defined.Considered as operators on the composite Hilbert space, B ⊗ F , the bosonand fermion creation and annihilation operators satisfy[a(†)b (x), a(†)f (y)]− = 0for all x, y ∈ X, where each “†” may be included or not.Proposition 2.2.4. For any n ∈ N, the norms of the creation and annihila-tion operators, a†(x) and a(x) for any x ∈ X, restricted to Dn(X), are‖a†(x)|Dn(X)‖ =√n+ 1 and ‖a(x)|Dn(X)‖ =√n.In particular, for bosons,‖a†b(x)|Bn(X)‖ =√n+ 1 and ‖ab(x)|Bn(X)‖ =√n.Lemma 2.2.5. For any n ∈ N and Y ∈ Xn/Sn, the results of acting on thebasis element δY with the boson creation, annihilation, and density operatorsat x ∈ X are:ab(x)δY =√µY (x) δY \{x},a†b(x)δY =√µY (x) + 1 δY unionsq{x}, andn(x)δY = µY (x)δY .Here, if (y1, . . . , yn) is a representative for Y , then Y unionsq {x} is the element ofXn+1/Sn+1 with representative (y1, . . . , yn, x). If x ∈ Y , for example, x = y1,then Y \{x} is the element of Xn−1/Sn−1 with representative (y2, . . . , yn). Ifx /∈ Y , then Y \{x} := Y . (However, in the latter case, µY (x) = 0, so theright side of the first equation is equal to zero anyway. Note that µY (x) wasdefined below Equation (2.12).)Similarly, for any Y ⊂ X, the fermion annihilation operator af (x) acts on28δY asaf (x)δY =(−1)k−1δY \{x} if yk = x for some k,0 otherwise,=n∑j=1δx(yj)(−1)j−1δY \{x}.The fermion creation operator at x, a†f (x) acts on δY asa†f (x)δY =(−1)k−1δY unionsq{x} if x /∈ Y and yk−1 < x < yk for some k,0 otherwise.Lemma 2.2.6. For any Y ∈ Xn/Sn with representative (y1, . . . , yn) ∈ Xn,the bosonic basis element δY can be obtained by repeated applications ofboson creation operators to the vacuum state:δY = cY a†b(y1) · · · a†b(yn)|0〉.For fermions, if Z = {z1, . . . , zn} ⊂ X with z1 < · · · < zn,δZ = δz1 ∧ · · · ∧ δzn = a†f (z1) · · · a†f (zn)|0〉.The bosonic case follows from Lemma 2.2.5 by induction.2.3 Quantum Statistical MechanicsWe will study quantum systems in thermodynamic equilibrium that are freeto exchange energy and particles with a heat bath. The set of possible statesis known as the grand canonical ensemble. The grand canonical ensemble isrepresented by a density matrix, defined in the following section.2.3.1 Density MatricesConsider a quantum system with a 2-dimensional Hilbert space H = C2,such as a spin-half particle. Let e1 = (1, 0) and e2 = (0, 1) and suppose29that p1, p2 ≥ 0 with p1 + p2 = 1. Suppose that the expected value of anyobservable A on H is〈A〉 = p1〈e1, Ae1〉+ p2〈e2, Ae2〉=Tr(ρA)Trρ,whereρ = p1Pe1 + p2Pe2 =p1 00 p2 ,and Pe1 and Pe2 are the projections onto e1 and e2 respectively. Then we saythat system is in a mixed state, or more precisely, the mixture of the purestate e1 with probability p1 and the pure state e2 with probability p2.In general, any positive, trace-class operator ρ on a Hilbert space H, maybe called a density operator and represents a statistical ensemble of purequantum states, also called a mixed state. In this context, an element ofthe Hilbert space (or the projective Hilbert space) is called a pure state. Asabove, if the system is in the mixed state ρ, the expectation value of anoperator A is〈A〉 =Tr(ρA)Trρ.2.3.2 Grand Canonical Partition FunctionFor a quantum system of many indistinguishable particles in thermodynamicequilibrium, the grand-canonical ensemble is represented by the density ma-trix,ρ = e−β(H−µN),provided that this operator is trace-class. Here, N is the particle-numberoperator as defined in Section 2.2.1; H is a self-adjoint operator on the Fockspace H called the Hamiltonian, which represents the total energy of thesystem. A typical example of such a Hamiltonian is given by a sum of theoperators dΓ(h) and dΓ(V ) from Section 2.2.1. The constants β and µ rep-resent the inverse temperature and chemical potential respectively.30The grand-canonical partition function is the trace of the density matrix,Z(β, µ) = Tr(e−β(H−µN)). (2.29)Aside from serving as a normalization factor for ρ, the partition function maybe modified and analyzed to extract a wealth of information about the quan-tum system, including the one and two-point correlation functions, whosebehaviour changes at the phase transitions described in the Introduction.Our approach to analyzing the partition function starts with expressing theoperator of the form H−µN in terms of creation and annihilation operatorsas outlined in Section 2.2.2, Equations (2.27) and (2.28). (This will be donein more detail for each system of interest in Chapters 4 and 5.) We then aimto expand the trace in the set of “eigenstates” of the annihilation operators.It will be through these coherent states that the fields and functional inte-grals over these fields will arise. In the next chapter, we will define coherentstates and explore their properties.31Chapter 3Coherent StatesThe goal of this chapter is to find eigenstates of the annihilation operators.The states that will arise are called coherent states and will be a critical toolin our construction of functional integral representations for quantum many-particle partition functions.Our review of coherent states here is similar to Negele and Orland [9], as wellas Feldman et. al. [10] in the case of bosons.As in the previous chapter, let X be a finite set with the counting measure.That is, for any function f : X → C,∫Xdx f(x) :=∑x∈Xf(x).3.1 Bosonic Coherent StatesWe set out to find an element of B(X) which is simultaneously an eigenvectorfor all ab(x) with x ∈ X. Consider first the simple case where |X| = 1. In thiscase, Bn(X) ∼= C for each n ∈ N0. Let en be the basis element correspondingto the n-boson space; that is, Bn(X) = Cen. For arbitrary φ ∈ B(X), thereis a sequence of complex numbers, {ϕn}∞n=0, such that∞∑n=0|ϕn|2 <∞32andφ =∞∑n=0ϕnen.Acting with the annihilation operator ab on the nth component,abϕnen = ϕn√nen−1.Supposing that φ is an eigenvector of ab, we choose the coefficients so thatϕn√n = αϕn−1 for some α ∈ C. If we choose ϕ0 = 1, it follows by induction(using ϕn√n = αϕn−1) thatϕn =αn√n!.Clearly, φ ∈ B(X). To see that φ ∈ D(ab),∞∑n=0‖abϕnen‖2 =∞∑n=1‖αϕn−1en−1‖2 = |α|2‖φ‖2.Hence,abφ =∞∑n=0abϕnen =∞∑n=1ϕn√nen−1 =∞∑n=1αϕn−1en−1 = αφ.We thus define the coherent state |α〉 to be this eigenvector of the annihilationoperator with eigenvalue α,|α〉 :=∞⊕n=0αn√n!en.For the remainder of this chapter, let X be any finite set with the countingmeasure. We generalize this definition of a coherent state to B(X), as follows.Definition 5 (Bosonic Coherent States). Consider a system of bosons withFock space B(X).(i) For any x ∈ X and α ∈ C,|αδx〉 :=∞∑n=0αn√n!δ⊗nsx =∞∑n=0αnn!a†b(x)n|0〉 = eαa†b(x)|0〉,33where |0〉 is the vacuum state, |0〉 = 1 ∈ C = B0 andδ⊗nsx = δx ⊗ · · · ⊗ δx︸ ︷︷ ︸n times.(ii) For any φ ∈ L2(X), the coherent state |φ〉 ∈ B(X) is|φ〉 :=⊗sy∈X|φ(y)δy〉 = e∫Xdy φ(y)a†b(y)|0〉. (3.1)In the following proposition, we note that after extending this definition ofcoherent states to the case |X| > 1, they continue to serve as eigenstates ofthe annihilation operators. Also, the creation operators act on the coherentstates |φ〉 as derivatives with respect to φ.Proposition 3.1.1. For any x ∈ X and φ ∈ L2(X),ab(x)|φ〉 = φ(x)|φ〉 and a†b(x)|φ〉 =∂∂φ(x)|φ〉. (3.2)Lemma 3.1.2. The coherent states may be expanded in the basis {δY :n ∈ N0, Y ∈ Xn/Sn} from Part (i) of Proposition 2.1.2, as follows. For anyφ ∈ L2(X),|φ〉 =∞∑n=0∑Y ∈Xn/Snφ(Y )cY δY ,where, if we think of Y ∈ Xn/Sn as an unordered list of elements y from X,φ(Y ) is the product of all those φ(y). To be more precise,φ(Y ) :=∏y∈Xφ(y)µY (y),where µY (y) is the number of appearances of y in Y . Also,cY =1√ ∏y∈XµY (y)!,as defined in Proposition 2.1.2.34Proposition 3.1.3. For all α, φ ∈ L2(X), the inner product of the coherentstates |α〉 and |φ〉 is〈α|φ〉 = e∫Xdy α(y)∗φ(y).Furthermore, the projection of |α〉 onto the n-boson subspace has norm‖P (n)|α〉‖ =‖α‖n√n!,where the norm on the right side is the norm of α in L2(X),‖α‖ =(∫Xdy |α(y)|2)1/2.We would also like to construct coherent states for fermions; however, thecanonical anti-commutation relations for fermion annihilation operators forceany complex eigenvalues to be zero. To see this, suppose that λ ∈ C isan eigenvalue for af (x) for some x ∈ X, with corresponding eigenvectorφ ∈ F(X). Thenaf (x)2φ = λ2φ.The left side is equal to 12 [af (x), af (x)]+, which is zero from Proposition 2.2.3.Therefore, λ = 0.However, if we work with a new algebra that includes anti-commuting el-ements, rather than the complex plane, we will see that it is possible toconstruct fermionic coherent states that will have similar properties to theirbosonic counterparts. We will define this algebra in the following section.3.2 The Grassmann AlgebraDefinition 6. The Grassmann algebra ∧V generated by the vector space V ,is defined to be the direct sum of all n-fold antisymmetric tensor products ofV :∧V :=dimV⊕n=0∧nV ,Note that, for n > dimV , ∧nV = {0}.35We will take V to be the 2|X|-dimensional vector space freely generated byBV := {φf (x) : x ∈ X} unionsq {φ∗f (x) : x ∈ X}and let G be the Grassmann algebra generated by V . Note that BV is a setof independent basis vectors indexed by x ∈ X and the presence or absenceof a star. To be clear, these are not complex variables and the star index isnot related to complex conjugation.In this section, it will be useful to enumerate all elements of BV so thatBV = {ξ1, . . . , ξD}, where D := dimV = 2|X|. Note that D is even. Theneach element of G can be expressed in the form,a(ξ) :=D∑n=0∑~i∈{1,...,D}n1≤i1<i2<···<in≤Da~i ξi1ξi2 · · · ξin , (3.3)where each a~i ∈ C. The coefficients for the sum of any two elements a(ξ)and b(ξ) are given by(a+ b)~i = a~i + b~i.The product of two elements of the Grassmann algebra (sometimes referredto as wedge product) is found by supplementing the usual distributive lawwith the commutation rule ξiξj = −ξjξi for all i, j.It will be useful to define an involution or adjoint operation on the Grassmannalgebra. For any element a(ξ) with the form above (3.3),a(ξ)∗ :=D∑n=0∑~i∈{1,...,D}n1≤i1<i2<···<in≤Da~i∗ ξ∗inξ∗in−1 · · · ξ∗i1 ,where, if ξj = φf (x) for some x ∈ X, then ξ∗j := φ∗f (x). On the other hand,if ξj = φ∗f (x) for some x ∈ X, then ξ∗j := φf (x). For the complex coefficient,a~i∗ is the complex conjugate of a~i. (Note the reversed order of the subscriptson the right-hand side.)Note that, for any a(ξ), b(ξ) ∈ G, a(ξ)∗∗ = a(ξ), (a(ξ)b(ξ))∗ = b(ξ)∗a(ξ)∗, and(αa(ξ) + βb(ξ))∗ = α∗a(ξ)∗ + β∗b(ξ)∗.36For convenience in notation, we’ll also leta(−ξ) :=D∑n=0∑~i∈{1,...,D}n1≤i1<i2<···<in≤Da~i (−ξi1)(−ξi2) · · · (−ξin)=D∑n=0∑~i∈{1,...,D}n1≤i1<i2<···<in≤D(−1)na~i ξi1ξi2 · · · ξin .An element of the Grassmann algebra a(ξ) is said to be even if it satisfiesa(−ξ) = a(ξ), and odd if it satisfies a(−ξ) = −a(ξ). We denote the setof all even elements of G by G+ and the set of odd elements by G−. ThenG = G+ ⊕ G− and we denote by P± the projection onto each subspace, G±.Even elements commute with all other elements of the Grassmann algebra.That is, for any a(ξ) ∈ G+ and any b(ξ) ∈ G,a(ξ)b(ξ) = b(ξ)a(ξ).On the other hand, odd elements a(ξ) ∈ G− satisfya(ξ)b(ξ) = b(−ξ)a(ξ).for all b(ξ) ∈ G. In general, for any a(ξ), b(ξ) ∈ G,a(ξ)b(ξ) = b(ξ)a+(ξ) + b(−ξ)a−(ξ)where a±(ξ) := P±a(ξ).We now define two linear maps on G that behave similarly to differentiationand integration in calculus.Definition 7 (Grassmann Differentiation and Integration).(i) For any i ∈ {1, . . . , D}, the Grassmann derivative,∂∂ξi: G → G37is the unique linear map which satisfies, for any element of the formξi1ξi2 · · · ξin where 1 ≤ i1 < i2 < · · · < in ≤ D,∂∂ξξi1ξi2 · · · ξin =(−1)j−1ξi1ξi2 · · · ξij−1ξij+1 · · · ξin if (∃j)ξij = ξ,0 otherwise.(ii) The Grassmann integral,∫dξD · · · dξ1 : G → Cis the unique linear map which is equal to zero on⊕D−1k=0 ∧kV and satisfiesthe property ∫dξD · · · dξ1 ξ1 · · · ξD = 1.Note that the Grassmann derivative is not a rate of change and the Grass-mann integral makes no reference to a measure. However, these maps satisfyproperties that mimic those of their namesakes, including a product rule andan integration-by-parts formula. For more details, see [19].Lemma 3.2.1. For any elements a(ξ), b(ξ) ∈ G,∫dξD . . . dξ1 a(ξ)b(ξ) =∫dξD . . . dξ1 b(−ξ)a(ξ) =∫dξD . . . dξ1 b(ξ)a(−ξ).Proof. From the definition of the Grassmann integral, only the terms of theintegrand of order D will contribute. Since D = 2|X| is even in this case, wemay project the integrand onto the even subspace G+.∫dξD . . . dξ1 a(ξ)b(ξ) =∫dξD . . . dξ1 P+a(ξ)b(ξ)=∫dξD . . . dξ1 [(P+a(ξ))(P+b(ξ)) + (P−a(ξ))(P−b(ξ))]=∫dξD . . . dξ1 [(P+b(−ξ))(P+a(ξ)) + (P−b(−ξ))(P−a(ξ))]=∫dξD . . . dξ1 P+(b(−ξ)a(ξ))=∫dξD . . . dξ1 b(−ξ)a(ξ).38In the third line, it is also true that (P−a(ξ))(P−b(ξ)) = (P−b(ξ))(P−a(−ξ)).Following the same steps, we obtain the second equality of the lemma.We now allow elements of the Grassmann algebra G to act as coefficients onthe fermionic Fock space F(X) by forming the tensor product between thetwo spaces. Then we will define a multiplication between elements of G andthe operators on F(X), which will be similar to the multiplication of twosuperalgebras [19]. This definition relies on splitting the space B(F(X)) oflinear operators on F(X) into even and odd parts. In order to do this, wewill require the following lemma.Lemma 3.2.2. Each linear operator on F(X) can be expressed as a linearcombination of products of creation and annihilation operators.Proof. Using the basis for F(X) from Equation (2.13), it suffices to showthat the claim holds for any operator of the form f 7→ δy1 ∧ · · · ∧ δyn〈δx1 ∧· · · ∧ δxm , f〉.δy1 ∧ · · · ∧ δyn〈δx1 ∧ · · · ∧ δxm , f〉 = a†f (y1) · · · a†f (yn)|0〉〈0|af (xm) · · · af (x1)fand since|0〉〈0| =∏y∈Xbaf (y)a†f (y),the claim follows.Let B+(F(X)) and B−(F(X)) be the subspaces of B(F(X)) generated byproducts of even and odd numbers of creation/annihilation operators (re-spectively). Then B(F(X)) = B+(F(X))⊕B−(F(X)).We define a multiplication operation on G ⊗B(F(X)) which is associative,distributive and satisfies the following property. For any A,B ∈ B(F(X))with A = A+ +A−, where A± ∈ B±(F(X)), and ψ, σ ∈ G with σ = σ+ +σ−,where σ± ∈ G±,[ψ ⊗ (A+ + A−)][(σ+ + σ−)⊗B]= ψσ+ ⊗ A+B + ψσ+ ⊗ A−B + ψσ− ⊗ A+B − ψσ− ⊗ A−B.The odd elements of the two algebras anticommute, while the even parts com-mute. This is the same as multiplication in the tensor product of two superal-39gebras [19], however, B(F(X)) is not a superalgebra since [af (x), a†f (y)]+ =δx(y) rather than 0.In particular, note that for any x, y ∈ X,φ(∗)f (x)⊗ a(†)f (y) = −[1⊗ a(†)f (y)][φ(∗)f (x)⊗ 1].This equation is intended to be read either by including or omitting all ofthe symbols in brackets. Put more casually,φ(∗)f (x)a(†)f (y) = −a(†)f (y)φ(∗)f (x).We will continue to use this casual notation, identifying an element σ ∈ Gwith σ⊗1 ∈ G ⊗B(F(X)), and A ∈ B(F(X)) with 1⊗A ∈ G ⊗B(F(X)).Likewise, we identify f ∈ F(X) with 1⊗ f ∈ G ⊗ F(X).We can now regard G ⊗ B(F(X)) as the space of linear operators fromG ⊗ F(X) to G ⊗ F(X). Furthermore, any linear operator A on G (such asGrassmann integration and differentiation, for example) can be consideredas an operator on G ⊗F(X) via A(a(ξ)⊗ f) := (Aa(ξ))⊗ f for any a(ξ) ∈ Gand f ∈ F(X).For any a(ξ), b(ξ) ∈ G and f, g ∈ F(X), we define the “inner product”,〈a(ξ)f, b(ξ)g〉 := a(ξ)∗b(ξ)〈f, g〉.This can be extended to all of G ⊗ F(X),〈∑iai(ξ)fi,∑jbj(ξ)gj〉:=∑i,jai(ξ)∗bj(ξ)〈fi, gj〉for all ai(ξ), bj(ξ) ∈ G and fi, gj ∈ F(X).403.3 Fermionic Coherent StatesAs in the beginning of the previous section (3.2), let G be the Grassmannalgebra over V , the vector space that is freely generated byBV = {φf (x) : x ∈ X} unionsq {φ∗f (x) : x ∈ X}.Definition 8. The fermionic coherent state is |φf〉 is defined to be|φf〉 := e−∫Xdx φf (x)a†f (x)|0〉. (3.4)Here, the exponential is defined on G ⊗B(F(X)) by its series expansion,eu =∞∑n=01n!un,for any u ∈ G ⊗B(F(X)). (Note that this sum must be finite, terminatingat some n ≤ |X|.)Since the terms φf (x)a†f (x) in the exponent of the fermionic coherent stateare “even” in G ⊗B(F(X)), they must commute with each other. Hence,e−∑x∈Xφf (x)a†f (x) =∏x∈Xe−φf (x)a†f (x).For each x ∈ X and k > 1, (−φf (x)a†f (x))k = 0. Hence, the series expansionof the exponential e−φf (x)a†f (x) terminates after the linear term. That is,e−φf (x)a†f (x) = 1− φf (x)a†f (x).Therefore,|φf〉 =∏x∈X(1− φf (x)a†f (x))|0〉.Again, since each factor (1−φf (x)a†f (x)) is even in G⊗B(F(X)), this prod-uct may be taken in any order.41On the other hand, we define〈φf | := 〈0|e−∑x∈Xaf (x)φ∗f (x) = 〈0|∏x∈X(1− af (x)φ∗f (x)).It is easy to see that, for any u ∈ G ⊗ F(X), 〈φf |u〉 is equal to the innerproduct of |φf〉 with u, as defined above.We are now prepared to achieve our goal for this section – to prove that, overthe Grassmann algebra, the fermionic coherent states behave as eigenvectorswith respect to the annihilation operators. Similarly, the fermion creationoperators act as Grassmann derivatives on the coherent states, much like theboson creation operators act as derivatives (with respect to the boson fields)on the boson coherent states.Proposition 3.3.1. For all x ∈ X,af (x)|φf〉 = φf (x)|φf〉 and a†f (x)|φf〉 = −∂∂φf (x)|φf〉.Likewise,〈φf |af (x) =∂∂φ∗f (x)〈φf | and 〈φf |a†f (x) = φ∗f (x)〈φf |.Proof.af (x)|φf〉 = af (x)∏y∈X(1− φf (y)a†f (y))|0〉=∏y 6=x(1− φf (y)a†f (y)) af (x)(1− φf (x)a†f (x))|0〉=∏y 6=x(1− φf (y)a†f (y))φf (x)|0〉=∏y 6=x(1− φf (y)a†f (y))φf (x)(1− φf (x)a†f (x))|0〉= φf (x)∏y∈X(1− φf (y)a†f (y))|0〉= φf (x)|φf〉.42Similarly,a†(x)|φf〉 = a†f (x)∏y(1− φf (y)a†f (y))|0〉=∏y 6=x(1− φf (y)a†f (y)) a†f (x)(1− φf (x)a†f (x))|0〉=∏y 6=x(1− φf (y)a†f (y)) a†f (x)|0〉= a†f (x)∏y 6=x(1− φf (y)a†f (y))|0〉= −∂∂φf (x)(1− φf (x)a†f (x))∏y 6=x(1− φf (y)a†f (y))|0〉= −∂∂φf (x)|φf〉.The proof is very similar for the dual-type equations in the second part ofthe proposition.3.4 Coherent States for Mixed SystemsFor a mixed system of P species of particles, bosons and/or fermions, a co-herent state is simply a tensor product of coherent states for the componentspecies. In this section, we define notation for the Hilbert spaces of thesemixed systems and their coherent states.Let Hp be the Fock space for the pth particle species. We choose the conven-tion of listing bosons first, so that the first Pb species are bosonic and thelast Pf species are fermionic, where Pb + Pf = P . We also assume that foreach p, there is a finite set Xp, which we call the “configuration space” forthe pth species, such that if p ≤ Pb, Hp = B(Xp) and if p > Pb, Hp = F(Xp).For example, if the pth species consists of spin-12 fermions on a lattice X, itsconfiguration space is Xp = X × Sp, where Sp = {↑, ↓}.43The Hilbert space for the full combined system is thenH := H1 ⊗H2 ⊗ · · · ⊗ HP= B(X1)⊗ · · · ⊗ B(XPb)⊗F(XPb+1)⊗ · · · ⊗ F(XP ).(3.5)The bosonic space isB := B(X1)⊗ · · · ⊗ B(XPb) (3.6)and the fermionic space isF := F(XPb+1)⊗ · · · ⊗ F(XP ). (3.7)It will be convenient to define, for any n = (n1, n2, . . . , nPb) ∈ NPb0 , the “n-boson subspace” Bn to be the subspace of B with np particles for each pthboson species,Bn := Bn1(X1)⊗ Bn2(X2)⊗ · · · ⊗ BnPb (XPb). (3.8)Taking the following direct sums, we obtain the subspace of B with at mostnp particles for each pth boson species,B≤n =( n1⊕i=0Bi(X1))⊗( n2⊕i=0Bi(X2))⊗ · · · ⊗nPb⊕i=0Bi(XPb) . (3.9)For any m = (m1, . . . ,mPb) and n = (n1, . . . , nPb) in NPb0 , if mp ≤ np for allp, we write “m ≤ n”. With this notation,B≤n =⊕m≤nBm.The corresponding subspaces of the full Hilbert space H areHn := Bn ⊗F and H≤n := B≤n ⊗F . (3.10)We denote the projections onto these subspaces by P (n) and Pn respectively,P (n) = PHn and Pn = PH≤n . (3.11)For p ≤ Pb, let φp ∈ L2(Xp). For p > Pb, let Gp be the Grassmann Algebra44over Vp, the vector space freely generated by{φp(x) : x ∈ Xp} unionsq {φ∗p(x) : x ∈ Xp}.The fields for the mixed systems are vector-valued with P components, onefor each species. We will represent these by bold Greek letters,φ := (φ1, . . . , φPb , φPb+1, . . . , φP ).The first Pb components are complex fields while the last Pf components areGrassmann fields. The multi-species coherent state |φ〉 is then defined to bethe tensor product,|φ〉 = |φ1〉 ⊗ |φ2〉 ⊗ · · · ⊗ |φP 〉.For such φ, we also define φ∗ := (φ∗1, φ∗2, . . . , φ∗P ). (For p ≤ Pb, φ∗p is the com-plex conjugate of φp ∈ L2(Xp), whereas for p > Pb, φ∗p is the ∗-Grassmannfield, representing the set of starred basis elements, φ∗p(x) for x ∈ Xp.)It will be useful to sometimes split up the bosonic and fermionic parts of thecoherent state and write|φ〉 = |φB〉 ⊗ |φF 〉,where |φB〉 = |φ1〉 ⊗ · · · ⊗ |φPb〉 and |φF 〉 = |φPb+1〉 ⊗ · · · ⊗ |φP 〉.For the dual map, we will often write,〈φ| = 〈φP | ⊗ 〈φP−1| ⊗ · · · ⊗ 〈φ1|.However, since the creation and annihilation operators corresponding to dif-ferent particle species commute, as do the corresponding Grassmann vari-ables, the order on the right side does not matter.The inner product between two coherent states for the mixed system is then〈α|φ〉 =P∏p=1〈αp|φp〉 =P∏p=1e∫Xpdx αp(x)∗φp(x).We will sometimes use the following shorthand notation for the right-hand45side,e∫dx α∗(x)·φ(x) :=P∏p=1e∫Xpdx αp(x)∗φp(x).3.5 Approximate Resolutions of the IdentityOur goal is to expand the partition function in the set of coherent states.Since〈φ|ψ〉 = e∫Xdx φ∗(x)ψ(x) 6= 0for all φ, ψ ∈ L2(X), the bosonic coherent states do not form an orthonormalbasis. However, they do span the whole bosonic Fock space B(X).In this section, we aim prove that the coherent states satisfy a type of “closurerelation” of the form,1 =∫ ∏x∈X[dφ∗(x)dφ(x)2pii]e−∫dy φ∗(y)φ(y)|φ〉〈φ|, (3.12)where the measuredφ∗(x)dφ(x)2i= dRe(φ(x))d Im(φ(x))represents the Lebesgue measure on C ∼= R2. The right-hand side is oftencalled a resolution of the identity.3.5.1 For BosonsFrom Proposition 3.1.3, for any φ ∈ L2(X), the norm of the operatore−∫dy |φ(y)|2|φ〉〈φ|is equal to 1. Since the integral in (3.12) runs over a domain of infinitemeasure, L2(X) ∼= C|X|, its convergence is unclear at first glance. We willmake a “large-field cutoff”, replacing the domain by a ball of radius R, andthen take the limit as R→∞.46Definition 9. For any R > 0, the measure µR on L2(X) is defined bydµR(φ∗, φ) :=∏x∈X[dφ∗(x)dφ(x)2piiχR(|φ(x)|)],where χR is the characteristic function of the interval [0, R]; that is,χR(x) =1 if 0 ≤ x ≤ R,0 otherwise,for all x ∈ R.Definition 10. For all r > 0, the bosonic approximate resolution of theidentity Ir is defined on B(X) byIrf :=∫L2(X)dµr(φ∗, φ)e−‖φ‖2|φ〉〈φ|f〉 (3.13)for all f ∈ B(X).In the following theorem, we outline some important properties of Ir. In par-ticular, Ir indeed converges to the identity operator (in the strong operatortopology) as r →∞. (This is essentially a copy of Theorem 2.26 from [10].)Theorem 3.5.1. For all r > 0, the bosonic approximate resolution of theidentity Ir satisfies the following properties.(i) Each element of the basis {δY : Y ∈ Xn/Sn} is an eigenvector of Irwith eigenvalueλr(Y ) =∏x∈XΓr(µY (x))µY (x)!, (3.14)whereΓr(s) :=∫ r20dt e−ttsfor all s > −1.(ii) The boson number operator Nb commutes with Ir.47(iii) The operator norm of Ir satisfies the bound‖Ir‖ ≤ 1.(iv) In the limit as r → ∞, Ir converges strongly to the identity operator.That is, for all f ∈ B(X),limr→∞‖Irf − f‖ = 0.(v) The difference between Ir and the identity operator 1 satisfies the fol-lowing bound:‖(1− Ir)Pn‖ ≤ |X|2n+1e−12 r2.Here, Pn is the orthogonal projection onto the subspace of at most nbosons,⊕nj=0 Bn(X).3.5.2 For FermionsIn the section, we use the notation from Section 3.3 and prove that fermionicanalogue of Equation (3.12) is exact.Lemma 3.5.2. The operator∫ (∏x∈Xdφ∗f (x)dφf (x))e−∫Xdx φ∗f (x)φf (x)|φf〉〈φf |is the identity operator on G ⊗ F(X). We call this operator a fermionicresolution of the identity. Here,∫X dx φ∗f (x)φf (x) =∑x∈X φ∗f (x)φf (x).Proof. For any Y ⊂ X with ordered elements y1 < · · · < yn, note that〈φf |δY 〉 = φ∗(y1) . . . φ∗(yn)and〈δY |φf〉 = φf (yn) · · ·φf (y1).48Let A be the proposed resolution of the identity,A :=∫ (∏x∈Xdφ∗f (x)dφf (x))e−∫Xdx φ∗f (x)φf (x)|φf〉〈φf |.Then, for any Y ⊂ X and Z ⊂ X with ordered elements y1 < · · · < yn andz1 < · · · < zm respectively, the matrix element 〈δZ , AδY 〉 is given by〈δZ , AδY 〉 =∫ (∏x∈Xdφ∗f (x)dφf (x))e−∫Xdx φ∗f (x)φf (x)〈δZ |φf〉〈φf |δY 〉=∫ (∏x∈Xdφ∗f (x)dφf (x))(∏x∈X(1− φ∗f (x)φf (x)))φf (zm) · · ·φf (z1)φ∗(y1) . . . φ∗(yn).If there is some i such that zi /∈ Y , then the integrand is of the form∏x 6=zifx(φf (x), φ∗f (x))(1− φ∗f (zi)φf (zi))φf (zi)=∏x 6=zifx(φf (x), φ∗f (x))φf (zi)where fx(φf (x), φ∗f (x)) is a polynomial in φf (x) and φ∗f (x). This expressioncontains no φ∗f (zi) and hence, the Grassmann integral∫ (∏x∈X dφ∗f (x)dφf (x))will map it to 0. Hence, if the integral is non-zero, Z ⊂ Y . We can use thesame reasoning to deduce that Y ⊂ Z if the integral is non-zero. Therefore,〈δZ , AδY 〉 = 0 unless Y = Z. In the case where Y = Z,〈δY , AδY 〉 =∫ (∏x∈Xdφ∗f (x)dφf (x))(∏x∈X(1− φ∗f (x)φf (x)))φf (yn) · · ·φf (y1)φ∗(y1) . . . φ∗(yn)=∫ (∏x∈Xdφ∗f (x)dφf (x))∏x/∈Y(1− φ∗f (x)φf (x))n∏i=1(1− φ∗f (yi)φf (yi))φf (yi)φ∗f (yi)49=∫ (∏x∈Xdφ∗f (x)dφf (x))∏x/∈Yφf (x)φ∗f (x)(n∏i=1φf (yi)φ∗f (yi))= 1This completes the proof.3.5.3 For Mixed SystemsAs in Section 3.4, we consider a mixed system of P species of particles. Recallthat the Hilbert space isH = H1 ⊗H2 ⊗ · · · ⊗ HP= B(X1)⊗ · · · ⊗ B(XPb)⊗F(XPb+1)⊗ · · · ⊗ F(XP ),where each Xp is a finite set. Here, Pb is the number of bosonic species, Pfis the number of fermionic species, and P = Pb + Pf .To each fermionic species (with p > Pb), we associate a Grassmann AlgebraGp over the vector space that is freely generated by{φp(x) : x ∈ Xp} unionsq {φ∗p(x) : x ∈ Xp}.Definition 11. For any r = (r1, . . . , rPb) ∈ (0,∞)Pb , let µbr be the productmeasure,µbr = µr1 × µr2 × · · · × µrPb .That is, µbr is the measure on the product space L2(X1)×· · ·×L2(XPb) givenbydµbr(φ∗B,φB) =Pb∏p=1dµrp(φ∗p, φp) =P∏p=1∏x∈Xp[dφ∗p(x)dφp(x)2piiχrp(|φp(x)|)]where φB := (φ1, . . . , φPb).We use a similar notation for the products of Grassmann integrals corre-sponding to the fermionic particle species. That is,∫dµf (φ∗F ,φF ) is the50map∫dµf (φ∗F ,φF ) =P∏p=Pb+1∫ ∏x∈Xpdφ∗p(x)dφp(x)where φF = (φPb+1, . . . , φP ).Combining the bosonic and fermionic parts, we let∫dµr(φ∗,φ) =∫dµbr(φ∗B,φB)∫dµf (φ∗F ,φF ).Definition 12. For any r ∈ (0,∞)Pb , the approximate resolution of theidentity Ir is the operator on H given byIr :=∫dµr(φ∗,φ)e−∫dx φ∗(x)·φ(x)|φ〉〈φ|, (3.15)where∫dx φ∗(x) ·φ(x) :=P∑p=1∫Xpdx φ∗p(x)φp(x).Note thatIr =Pb⊗p=1Ip,rp⊗ 1F ,where Ip,rp is the approximate resolution of the identity on B(Xp),Ip,rp =∫dµrp(φ∗, φ)e−∫Xpφ∗(x)φ(x)dx|φ〉〈φ|.Theorem 3.5.3. The approximate resolution of the identity Ir satisfies thefollowing properties.(i) For all n = (n1, . . . , nPb) ∈ NPb0 and all Y = (Y1, . . . , YPb) ∈ Xn11 /Sn1 ×· · · ×XnPbP /SnPb ,δY := δY1 ⊗ · · · ⊗ δYPbis an eigenvector of Ir. That is,IrδY = λr(Y)δY,51with eigenvalueλr(Y) :=Pb∏p=1λrp(Yp),where λrp(Yp) is the eigenvalue of Ip,rp corresponding to δYp , as definedin (3.14).(ii) For each p, Ir commutes with the pth-species particle number operator,Np :=∫Xpdx a†p(x)ap(x),where, for all x ∈ Xp, a†p(x) is the creation operator and ap(x) is theannihilation operator for the pth particle species. That is,a(†)p (x) =a(†)b (x) if x ≤ Pb,a(†)f (x) if x > Pb,where instances of “(†)” may be all included (and read as “†”) or allexcluded.(iii) For all r ∈ (0,∞)Pb , ‖Ir‖ ≤ 1, and Ir converges strongly to the identityoperator on H as min1≤p≤Pbrp →∞.(iv) For all n ∈ NPb0 and r ∈ (0,∞)Pb ,‖(1− Ir)Pn‖ ≤Pb∑p=1|Xp|2np+1e−r2p/2,where Pn is the projection onto H≤n as defined in Equation (3.11).Theorem 3.5.3 is an extension of its single-species boson analogue, Theorem2.26 in [10]. Parts (ii) and (iii) follow immediately from the definition of Irand Theorem 2.26 (b and c) in [10]. Part (iv) may be obtained by provingthe inequality for simple tensors and then extending linearly to H≤n. Thedifference between f1⊗ · · · ⊗ fPb and I1,r1f1⊗ · · · ⊗ IPb,rPbfPb may be writtenas a telescoping sum, so that we can apply the bound from Theorem 2.26 (d)in [10] for each species of bosons.523.6 Trace FormulaWhen attempting to expand the partition function,Z(µ, β) = Tr(e−β(H−µN)),using coherent states, we might intuitively start by taking inner products ofthe form〈φ|e−β(H−µN)|φ〉,then dividing by 〈φ|φ〉 and integrating over all φ. Roughly speaking, wemight expect a formula that looks likeTr(e−β(H−µN)) =∫dφ∗dφ e−∫Xdx φ∗(x)φ(x)〈φ|e−β(H−µN)|φ〉.In this section, we will show that this approach essentially works, providedthat we take an appropriate large-field cutoff for bosons.3.6.1 For BosonsFor bosons, we quote Proposition 2.28 (a) from [10], except that we do notrequire the operator B to commute with the boson number operator. (Theproof given in [10] does not actually require this condition.)Proposition 3.6.1. Let X be a finite set and let B be a bounded operatoron B(X). For all r > 0, BIr is trace class andTrBIr =∫dµr(φ∗, φ) e−∫Xdy |φ(y)|2〈φ|B|φ〉.3.6.2 For FermionsFor fermions, the intuitive formula is exactly correct, except for a minus sign.The integral in this case is a Grassmann integral.Proposition 3.6.2. Let X be a finite set. For any linear operator B on thefermionic Fock space F(X),Tr B =∫ ∏x∈Xdφ∗f (x)dφf (x) e−∫Xdx φ∗f (x)φf (x)〈−φf |B|φf〉,53where〈−φf | = 〈0|e∑x∈Xaf (x)φ∗f (x) = 〈0|∏x∈X(1 + af (x)φ∗f (x)).Proof. Let {uj}2|X|j=1 be an orthonormal basis for F(X). Expanding the traceof B in this basis,Tr B =2|X|∑j=1〈uj, Buj〉.Then inserting the fermionic resolution of the identity from Lemma 3.5.2 intoeach inner product,Tr B =2|X|∑j=1〈uj, B∫ (∏x∈Xdφ∗f (x)dφf (x))e−∫Xdx φ∗f (x)φf (x)|φf〉〈φf |uj〉=∫ (∏x∈Xdφ∗f (x)dφf (x))e−∫Xdx φ∗f (x)φf (x)2|X|∑j=1〈uj, B|φf〉〈φf |uj〉=∫ (∏x∈Xdφ∗f (x)dφf (x))e−∫Xdx φ∗f (x)φf (x)2|X|∑j=1〈−φf |uj〉〈uj, B|φf〉=∫ (∏x∈Xdφ∗f (x)dφf (x))e−∫Xdx φ∗f (x)φf (x)〈−φf |B|φf〉.In the third line, we’ve applied Lemma 3.2.1.3.6.3 For Mixed SystemsFor mixtures of P species of bosons and/or fermions, we use the notationfrom Sections 3.4 and 3.5.3.Proposition 3.6.3. Suppose that B is a bounded operator on H. Then forall r ∈ (0,∞)Pb , BIr is trace class andTr BIr =∫dµr(φ∗,φ) e−∫dy φ∗(y)·φ(y)〈φ˜|B|φ〉,54where |φ˜〉 = |φB〉 ⊗ | −φF 〉 and∫dy φ∗(y) ·φ(y) :=P∑p=1∫Xpdy φ∗p(y)φp(y).Proof. Since B is bounded and Ir is trace-class, BIr is trace class.For each n = (n1, n2, . . . , nPb) ∈ NPb0 , letXn/Sn := Xn11 /Sn1 ×Xn22 /Sn2 × · · · ×XnPbPb /SnPb . (3.16)Furthermore, for each Y = (Y1, Y2, . . . , YPb) ∈ Xn/Sn, letδY := δY1 ⊗ δY2 ⊗ · · · ⊗ δYPb .Then {δY : Y ∈ Xm/Sm,m ≤ n} is an orthonormal basis for B≤n.Let {fk : k = 1, . . . , dimF} be any orthonormal basis for F . (Note thatdimF = 2∑Pp=Pb+1|Xp|.)Expanding the trace of BIr in the basis {δY ⊗ fk},Tr(BIr) =dimF∑k=1∑n∈NPb0∑Y∈Xn/Sn〈δY ⊗ fk, BIr δY ⊗ fk〉 (3.17)=dimF∑k=1∑n∈NPb0∑Y∈Xn/Snλr(Y)〈δY ⊗ fk, B δY ⊗ fk〉. (3.18)Since λr(Y) satisfies the bound,maxY∈Xn/Sn∣∣∣λr(Y)∣∣∣ ≤Pb∏p=1maxYp∈Xnpp /Snp|λrp(Yp)| ≤Pb∏p=1(r2p|Xp|)npnp!,and dimHn ≤ 2∑Pp=Pb+1|Xp|∏Pbp=1|Xp|np , the sum in (3.18) is absolutely con-55vergent; that is,dimF∑k=1∑n∈NPb0∑Y∈Xn/Sn∣∣∣λr(Y)〈δY ⊗ fk, B δY ⊗ fk〉∣∣∣≤ 2∑Pp=Pb+1|Xp|‖B‖Pb∏p=1∞∑np=0(rp|Xp|)2npnp! <∞.Hence, Tr(BIrPn) converges to Tr(BIr) as each np →∞.By definition of Ir,Tr(BIrPn)=∑m≤n∑Y∈Xm/Sm∫dµr(φ∗,φ) e−∫dy φ∗(y)·φ(y)〈δY, B|φ〉〈φ|δY〉=∑m≤n∑Y∈Xm/Sm∫dµr(φ∗,φ) e−∫dy φ∗(y)·φ(y)〈φ˜|δY〉〈δY, B|φ〉=∫dµr(φ∗,φ) e−∫dy φ∗(y)·φ(y)〈φ˜|PnB|φ〉.The integrand is bounded by ‖B‖ for all n. By the dominated convergencetheorem, the right-hand side converges to∫dµr(φ∗,φ) e−∫dy φ∗(y)·φ(y)〈φ˜|B|φ〉.as min1≤p≤Pbnp →∞.56Chapter 4Particle-Number-PreservingInteractionsOur goal is to construct a functional integral representation for the partitionfunction of a gas consisting of an arbitrary finite number of particle species(bosons and/or fermions) interacting on a finite lattice, with a Hamiltonianthat preserves the total number of particles in each species.4.1 The Physical Setting and NotationThe first step of this project is to create notation that is compact, intuitive,and general enough to allow for the treatment of any finite number of bosonicand/or fermionic particle species. We review and build upon some of thenotation that has already been defined in Chapter 3.4.1.1 The Hilbert SpaceRecall from Section 3.4 that the Hilbert space for a mixed gas with P speciesof particles isH = H1 ⊗H2 ⊗ · · · ⊗ HP ,57where Hp is the Fock space for the pth species. Listing the bosons beforefermions,Hp =B(Xp) if p ≤ PbF(Xp) if p > Pb,where Xp is the configuration space for the pth species of particles and Pb isthe number of bosonic species. We also let Pf be the number of fermionicspecies so that P = Pb + Pf .The bosonic and fermionic spaces areB = B(X1)⊗ · · · ⊗ B(XPb) and F = F(XPb+1)⊗ · · · ⊗ F(XP )respectively.For each n = (n1, n2, . . . , nPb) ∈ NPb0 ,Bn = Bn1(X1)⊗ Bn2(X2)⊗ · · · ⊗ BnPb (XPb).andB≤n =( n1⊕i=0Bi(X1))⊗( n2⊕i=0Bi(X2))⊗ · · · ⊗nPb⊕i=0Bi(XPb) .The corresponding subspaces of H areHn = Bn ⊗F and H≤n = B≤n ⊗F .We denote the projections onto these subspaces by P (n) and Pn respectively,P (n) = PHn and Pn = PH≤n .4.1.2 Notation for Particle Coordinates and FieldsFor each p, we will typically use the lower-case letters x and y to denoteelements of Xp. For any np ∈ N, a vector of np elements of Xp will be58indicated with an arrow,~xp = (xp1 , xp2 , . . . , xpnp ) ∈ Xnpp .Furthermore, these elements of Xnpp will be collected to form the (nested)vector,x := (~x1, ~x2, . . . , ~xP ) ∈ Xn11 ×Xn22 × · · · ×XnPP .To condense the notation on the right-hand side, if n := (n1, . . . , nP ), letXn := Xn11 × · · · ×XnPP . (4.1)The union of all such elements isX :=⋃n∈NP0Xn. (4.2)For any element x of X, let n(x) = (n1(x), . . . , nP (x)) be the element n ofNP0 such that x ∈ Xn.We will denote the `1 norm of an element n ∈ NP0 by |n|. That is,|n| =P∑p=1np. (4.3)Hence, for any x ∈ X, then |n(x)| =∑Pp=1 np(x) is the total number of par-ticles whose positions are represented by x.The concatenation of two elements x = (~x1, ~x2, . . . , ~xP ) ∈ Xn and y =(~y1, ~y2, . . . , ~yP ) ∈ Xm is defined to bex ◦ y = (~x1 ◦ ~y1, ~x2 ◦ ~y2, . . . , ~xP ◦ ~yP ) ∈ Xn+m, (4.4)where, for each p,~xp ◦ ~yp = (xp1 , xp2 , . . . , xpnp , yp1 , yp2 , . . . , ypmp ). (4.5)Given an element x ∈ X with n(x) = n, we can writex = (~x1, · · · , ~xP ) = (x11 , . . . , x1n1 , · · · , xP1 , . . . , xPnP )59Consider the subscripts of each x as an ordered pair (i, j) so thatx(i,j) := xij .Let Ix be the set of all indices for components of x,Ix := {(1, 1), . . . , (1, n1), (2, 1) . . . , (2, n2), . . . , (P, 1), . . . , (P, nP )}. (4.6)There are |n(x)| distinct elements in this set. We define an order on theseelements, just as they are listed above in (4.6). That is,(p, n) < (q,m)whenever p < q, or in the case p = q, whenever n < m.For any subset U of Ix, xU is defined to be the vector of components of xwhose indices lie in U , with the order of their indices preserved. In otherwords, the components of xU form the set {x(p,n) : (p, n) ∈ U}, and they areordered by increasing (p, n).For U ⊂ Ix, we also let np(U) be the number of coordinates in U representingparticles of the pth species. That is,np(U) := |{m ∈ N : (p,m) ∈ U}| . (4.7)Since we will often want to count only the coordinates corresponding tofermions, we also letn(f)p (U) := np(U)χ[Pb+1,P ](p), (4.8)where χ[Pb+1,P ] is the characteristic function of the interval [Pb + 1, P ]. Thatis, χ[Pb+1,P ](p) is equal to 1 when the pth species is fermionic, and 0 otherwise.For any p ∈ {1, . . . , P}, m,n ∈ N with m ≤ n, and ~x ∈ Xnp , let~x[≤m] := (x1, . . . , xm) (4.9)and~x[>m] := (xm+1, . . . , xn).60Similarly, for any m,n ∈ NP0 with m ≤ n, and any x ∈ Xn,x[≤m] = (~x1[≤m1] , . . . , ~xP[≤mP ]) (4.10)andx[>m] = (~x1[>m1] , . . . , ~xP[>mP ]). (4.11)It will be useful to consider partitions of the set Ix. A partition P of Ix isa collection of nonempty disjoint subsets U of Ix whose union is equal toIx. For each U ∈ P , let ~U be the vector formed by the components of Uin increasing order. We then order the elements of P by increasing largestelement, so thatIx = U1 unionsq U2 unionsq · · · unionsq U|P|−1 unionsq U|P|,where U|P| contains the largest element (P, nP ) of Ix, U|P|−1 contains thelargest element of Ix\U|P|, and so on. Then the sign of the partition P ,(−1)P , is defined to be the sign of the permutation required to bring thecoordinates of each fermionic species from the concatenated list~U1 ◦ · · · ◦ ~U|P|back to their order in Ix.Recall that the fields for the mixed gas are of the formα = (α1, . . . , αPb , αPb+1, . . . , αP )where the first Pb components are complex fields and the last Pf componentsare Grassmann fields. For a given x ∈ X1 × · · · ×XP , α∗(x) is defined to bethe productα∗(x) := α∗1(~x1) · · ·α∗P (~xP ), (4.12)where for each p,α∗p(~xp) := α∗p(xp1)α∗p(xp2) · · ·α∗p(xpnp ). (4.13)We similarly define a differential operator,∂∂α∗(x):=∂∂α∗P (~xP )· · ·∂∂α∗1(~x1), (4.14)61where, for each p,∂∂α∗p(~xp):=∂∂α∗p(xp,np)∂∂α∗p(xp,np−1)· · ·∂∂α∗p(xp,1). (4.15)Note that the order of the position coordinates are decreasing in this prod-uct, whereas they are increasing in Equation (4.13).For any n = (n1, n2, . . . , nP ) ∈ NP0 , we use the multi-index notation,n! := n1!n2! · · ·nP !. (4.16)We also defineSn := Sn1 × Sn2 × · · · × SnP . (4.17)The sign of σ = (σ1, . . . , σP ) ∈ Sn is(−1)σ :=P∏p=Pb+1(−1)σp . (4.18)We may let any vector of permutations σ of Sn act on Xn viaσ : Xn → Xn(~x1, . . . , ~xP ) 7→ (σ1(~x1), . . . , σP (~xP )). (4.19)(To be clear, each ~xp ∈ Xnpp in the second line.) Likewise, we can allow σ toact on functions f with domain Xn as(σf)(x) = f ◦ σ−1(x) = f(σ−1(x)),where σ−1 = (σ−11 , σ−12 , . . . , σ−1P ).The symmetrization operator S(n) is defined asS(n) :=1n!∑σ∈Sn(−1)σσ. (4.20)62Note that this is also equal toS(n) =Pb⊗p=1S(np)b (Xp)⊗P⊗p=Pb+1S(np)f (Xp).In instances where a function has several arguments in Xn, we will use asubscript to indicate which argument is being symmetrized, as inS(n)x f(x,y) =1n!∑σ∈Sn(−1)σf(σ−1(x),y).4.1.3 Creation and Annihilation OperatorsFor each pth species, creation and annihilation operators at x ∈ Xp are de-noted by a†p(x) and ap(x), respectively.For any p and ~x = (x1, x2, . . . , xn) ∈ Xnp , leta†p(~x) := a†p(x1)a†p(x2) · · · a†p(xn) and ap(~x) := ap(xn)ap(xn−1) · · · ap(x1).(Note that the order of the annihilation operators is reversed in the secondequation above.)Furthermore, for any x = (~x1, ~x2, . . . , ~xP ) ∈ X, we definea†(x) := a†1(~x1)⊗ · · · ⊗ a†P (~xP ) and a(x) := a1(~x1)⊗ · · · ⊗ aP (~xP ).The number operator (densely defined for bosons) on Hp for the pth speciesisNp =∫Xpdx a†p(x)ap(x).4.1.4 The HamiltonianThe Hamiltonian H is a linear operator on the Hilbert space H that repre-sents the total energy of the system. Generalizing Proposition 2.14 of [10],63we write H in the form,H =∫X2dx dy a†(x)h(x,y)a(y),where h : X2 → C is the kernel of H. For example, if H includes the kineticenergy of each particle as well as a two-body interaction, h would be of theformh(x,y) =hp(x, y) if x = x ∈ Xp and y = y ∈ Xp,v(x1, x2)δx1(y1)δx2(y2) if n(x) = n(y) and |n(x)| = 2,0 otherwise.(4.21)Note that h(x,y) depends on the sizes of its arguments, n(x) and n(y), aswell as their values.We make four main assumptions about the Hamiltonian H in this chapter:1. The kernel satisfies h(y,x) = h∗(x,y) for all x,y ∈ X;2. H preserves the number of particles in each species;3. The order M of H is finite and strictly greater than 1, whereM := max{max(|n(x)|, |n(y)|) : x,y ∈ X and h(x,y) 6= 0}.4. H satisfies the boundsPb∑p=1(AN2p − B˜Np)≤ H ≤ C˜Pb∏p=1NMp , (4.22)where A > 0, B˜, and C˜ are real constants that may depend on M andeach |Xp|.Note that the second assumption implies that h(x,y) is only nonzero whenn(x) = n(y).64In [10], the bounds of Equation (4.22) for one species of bosons with aM = 2-body interaction were not assumed directly, but rather deduced from proper-ties assumed of the kernel h. In particular, the lower bound follows from theassumption that the second-order interaction is repulsive, just as particlesof like charge repel under the Coulomb interaction. In particular, repulsive2-body interactions lead to lower bounds on H that are quadratic in theparticle number, as in (4.22); see Definition 2.16 and Proposition 2.17 of[10]. However, if we allow higher-order interactions, we could also deducethis bound by assuming that the term of highest order in each boson speciesis repulsive, in some sense. For example, for anyM ∈ N, consider one speciesof bosons with an M th-order interaction of the formV =∫XMdx1 · · · dxM a†(x1) · · · a†(xM)v(x1, · · · , xM)a(x1) · · · a(xM),where v : XM → R. We assume that V is repulsive in the sense thatv0 := inf{∫XMdx1 · · · dxM v(x1, · · · , xM)ρ(x1) · · · ρ(xM)∣∣∣∣∫dx ρ(x)M = 1, ρ ≥ 0}is (strictly) positive. Then for any n ∈ N and Y ∈ Xn/Sn,〈δY , V δY 〉 =∫XMdx1 · · · dxM v(x1, · · · , xM)〈δY , n(x1) · · ·n(xM)δY 〉=∫XMdx1 · · · dxM v(x1, · · · , xM)µY (x1) · · ·µY (xM)≥ v0∫Xdx µY (x)M≥ v0 maxx∈XµY (x)M≥ v0(∫X dx µY (x)|X|)M=v0|X|MnM .Since the set {δY : Y ∈ Xn/Sn} forms an orthonormal basis of eigenvectorsof V , it follows thatV ≥v0|X|MNM ,65where N is the boson number operator. Of course, NM ≥ N2; hence, thelower bound of the form (4.22) indeed applies to V . We could perform a sim-ilar analysis for other higher-order interactions, both for the lower and upperbounds. However, we will simply assume the bounds in Equation (4.22),rather than assuming conditions on h.Since the creation and annihilation operators satisfy (anti)commutation re-lations, the kernel h must have symmetry properties as well. That is, for anyσ,pi ∈ Sn and x,y ∈ Xn,h(σ(x),pi(y)) = (−1)σ(−1)pih(x,y). (4.23)(To review the notation above, please see page 62.)Under these assumptions, it is easy to see that H is essentially self-adjointon the subspace of H with a finite number of bosons,H(finite) = B(finite)(X1)⊗ B(finite)(X2)⊗ · · · ⊗ B(finite)(XPb)⊗F ,whereB(finite)(X) := {f ∈ B(X) : (∃m ∈ N)(∀n > m)P (n)f = 0}.(Note that m depends on f .) Alternately, B(finite)(X) = SbD(finite)(X), asdefined in Chapter 2.Recall from Equation (2.29) of Section 2.3.2 that the partition function fora single species of indistinguishable particles isTr(e−β(H−µN)),where β is the inverse temperature, H is the Hamiltonian, µ is the chemicalpotential, and N is the particle number operator.With p different species of particles, we replace µN with the sum∑Pp=1 µpNp,where µp is the chemical potential for the pth particle species. For conve-nience, we form the vector of all P chemical potentials,µ := (µ1, . . . , µP ),66as well as the vector of all P particle number operators,N := (N1, . . . , NP ).Then the partition function may be written asZ(β,µ) = Tr e−β(H−µ·N).Since our main goal is to find a functional integral representation for thepartition function, we will be more commonly working with the operator inthe exponent rather than the Hamiltonian H. We will denote this self-adjointoperator by K,K := H − µ ·N.Since H and µ ·N each preserve the number of particles in each species, sotoo must K. Provided that the order of H is nonzero, H and K also havethe same order.Let k : X2 → C be the kernel of K, such thatK =∫X2dx dy a†(x)k(x,y)a(y).Assumptions 1-4 above on H also carry over for K. That is,1. The kernel satisfies k(y,x) = k∗(x,y) for all x,y ∈ X;2. K preserves the number of particles in each species;3. The order M of K is finite, whereM := max{max(|n(x)|, |n(y)|) : x,y ∈ X and k(x,y) 6= 0}.4. K is stable in the sense that it is bounded from below. In particular,Pb∑p=1(AN2p −BNp)≤ K ≤ CPb∏p=1NMp (4.24)where A > 0 and B are constants and Np is the total boson numberoperator for the pth species.67In particular, note that Equation (4.24) implies that K is bounded below bya constant kL,K ≥ kL, where kL :=−PbB24A. (4.25)4.2 A Functional Integral Representationfor the Partition FunctionThe main goal of this section is to prove the coherent-state functional integralrepresentation for the partition function in the following theorem.Theorem 4.2.1. Let R be a map from N to (0,∞)Pb satisfyinglimq→∞qe−Rp(q)2/2 = 0for each p ∈ {1, . . . , Pb}. ThenTr e−βK = limq→∞∫ ∏τ∈Tq[dµR(q)(φ∗τ ,φτ )e−∫dy φ∗τ (y)·φτ (y)] ∏τ∈Tq〈φτ−|e−K |φτ 〉(4.26)where Tq := {, 2, . . . , q},  :=βq ,φ0 := φ˜β = (φ01 , . . . , φ0Pb ,−φ0Pb+1 , . . . ,−φ0P ),and∫dy φ∗τ (y) ·φτ (y) :=P∑p=1∫Xpdy φ∗τp(y)φτp(y).The lemma below will be required to prove this theorem.Lemma 4.2.2. Let R be a map from N to (0,∞)Pb satisfyinglimq→∞qe−Rp(q)2/2 = 0for each p ∈ {1, . . . , Pb}. Then,Tr e−βK = limq→∞Tr(e−βqKIR(q))q−1e−βqK68and for any φ,φ′,〈φ|e−βK |φ′〉 = limq→∞〈φ|(e−βqKIR(q))q−1e−βqK |φ′〉.Proof. This proof is very similar to that of Lemma 3.4 in [10]. As in [10], weintroduce the notationAi :=e−βqK if i is oddIR(q) if i is evenand Bi :=e−βqK if i is odd1 if i is evenso that(e−βqKIR(q))q−1e−βqK =2q−1∏i=1Ai and e−βK =2q−1∏i=1Bi.Let n := (n, n, . . . , n) ∈ NPb for n ∈ N. Then Pn is the projection ontoH≤n = B≤n ⊗F =(n⊕i=0Bi(X1))⊗(n⊕i=0Bi(X2))⊗ · · · ⊗(n⊕i=0Bi(XPb))⊗F .By the triangle inequality for the trace norm ‖ · ‖1 = Tr | · |,Tr∣∣∣∣∣∏iAi −∏iBi∣∣∣∣∣≤ Tr∣∣∣∣∣(∏iAi −∏iBi)Pn∣∣∣∣∣+ Tr∣∣∣∣∣∏iAi(1− Pn)∣∣∣∣∣+ Tr∣∣∣∣∣∏iBi(1− Pn)∣∣∣∣∣. (4.27)We start by bounding the first term, which can be written as a telescopingsum,Tr∣∣∣∣∣(∏iAi −∏iBi)Pn∣∣∣∣∣= Tr∣∣∣∣∣∣2q−1∑j=1j−1∏i=1Ai (Aj −Bj)2q−1∏i=j+1Bi Pn∣∣∣∣∣∣≤ Tr(Pn)2q−1∑j=1∥∥∥∥∥∥j−1∏i=1Ai (Aj −Bj)2q−1∏i=j+1Bi Pn∥∥∥∥∥∥.69Note that‖Ai‖, ‖Bi‖ ≤eβq kL if i is odd,1 if i is even,where kL ∈ R is the lower bound for K from Equation (4.25).Since Ai −Bi = IR(q) − 1 when i is even and is zero otherwise,Tr∣∣∣(∏Ai −∏Bi)Pn∣∣∣ ≤ Tr(Pn)(q − 1)eβkL‖(IR(q) − 1)Pn‖≤ Tr(Pn)(q − 1)eβkLPb∑p=1|Xp|2np+1e−Rp(q)2/2, (4.28)where the bound on ‖(IR(q)−1)Pn‖ in the second line is from Theorem 3.5.3.The large-field cutoffs Rp(q) have been chosen to grow quickly enough so thatthe bound in (4.28) shrinks to 0 as q →∞, for any fixed n ∈ N.Using the bound from Equation (4.24), the other two terms in (4.27) satisfyTr∣∣∣∏Ai(1− Pn)∣∣∣ ,Tr∣∣∣∏Bi(1− Pn)∣∣∣ ≤∑m∈NPb(∃p)mp>ne−β(Am2−B|m|) Tr(P(m)),(4.29)where m2 =∑Pbp=1m2p and |m| =∑Pbp=1mp. The trace of P(m) is equal to thedimension of Hm = Bm ⊗F ,dim(Hm) =Pb∏p=1dim(Bmp(Xp))P∏p=Pb+12|Xp|.The dimension of the mp-boson space, Bmp(Xp) isdim(Bmp(Xp)) =(mp + |Xp| − 1)!mp!(|Xp| − 1)!≤(mp + |Xp| − 1)|Xp|−1(|Xp| − 1)!70≤|Xp||Xp|−1(|Xp| − 1)!m|Xp|−1p .Hence,Tr(Pm) ≤ CPb∏p=1m|Xp|−1pwhere C =∏Pbp=1|Xp||Xp|−1(|Xp|−1)!∏Pp=Pb+1 2|Xp| is a constant that depends on the sizeof each Xp (but not on q or mp).Applying this to Equation (4.29),Tr∣∣∣∏Ai(1− Pn)∣∣∣ ,Tr∣∣∣∏Bi(1− Pn)∣∣∣≤∑m∈NPb(∃p)mp>ne−β(Am2−B|m|)CPb∏p=1m|Xp|−1p≤ C∑m∈NPb(∃p)mp>nPb∏p=1e−β(Am2p−Bmp)m|Xp|−1p≤ C(∞∑m=0e−β(Am2−Bm)mMX)Pb−1 ∞∑m=ne−β(Am2−Bm)mMX→ 0 as n→∞,where MX := max1≤p≤Pb|Xp| − 1. Hence, the bound in (4.29) can be madearbitrarily small, as long as n is chosen to be sufficiently large (independentlyof q). It follows that∏Ai −∏Bi → 0 as q → ∞ in the trace norm, andhence also in operator norm, as well as weakly. The claim follows.Proof of Theorem 4.2.1. We start by expressing the partition function as asum over all n-particle spaces,Tr e−βK =∑n∈NPb0TrHn(e−βKP (n)).Since each Hn is finite-dimensional, IR(q) maps each Hn to itself, and IR(q)71converges strongly to 1 as q →∞ (from Theorem 3.5.3), it follows thatTr e−βK =∑n∈NPb0limq→∞TrHn(e−βKP (n)IR(q)).Note that∣∣∣TrHn(e−βKIR(q))∣∣∣ ≤ dim(Hn)‖e−βKP(n)‖ ≤ dim(Hn)e−β(An2−B|n|),which is summable over n, as we’ve just seen in the proof of the Lemma5.3.9. Hence, by the dominated convergence theorem,Tr e−βK = limq→∞Tr(e−βKIR(q)).From Proposition 3.6.3, it follows thatTr e−βK = limq→∞∫dµR(q)(φ∗0,φ0)e−∫dy φ∗0(y)·φ0(y)〈φ˜0|e−βK |φ0〉Comparing this with the right-hand side of Equation (4.26),∫dµR(q)(φ∗0,φ0)e−∫dy φ∗0(y)·φ0(y)〈φ˜0|e−βK |φ0〉−∫ ∏τ∈Ts[dµR(q)(φ∗τ ,φτ )e−∫dy φ∗τ (y)·φτ (y)] ∏τ∈Ts〈φ˜τ−|e−K |φτ 〉=∫dµR(q)(φ∗0,φ0)e−∫dy φ∗0(y)·φ0(y)〈φ˜0|e−βK |φ0〉−∫dµR(q)(φ∗0,φ0)e−∫dy φ∗0(y)·φ0(y)〈φ˜0|(e−βqKIR(q))q−1e−βqK |φ0〉=∫dµR(q)(φ∗0,φ0)e−∫dy φ∗0(y)·φ0(y)〈φ˜0|e−βK −(e−βqKIR(q))q−1e−βqK |φ0〉=∫ Pb∏p=1∏x∈Xp[dφ∗p(x)dφp(x)2piiχRp(q)(|φp(x)|)]e−∫dy φ∗0B (y)·φ0B (y)〈φ0B |TrF[e−βK −(e−βqKIR(q))q−1e−βqK]|φ0B〉, (4.30)where φ0B is the bosonic component of φ0,φ0B := (φ01 , . . . , φ0Pb ).72From Lemma 4.2.2, we know that the integrand converges to 0 as q → ∞.In order to apply the dominated convergence theorem again, note that forany q,∣∣∣〈φ0B |TrF[e−βK]|φ0B〉∣∣∣ ≤ dimF∑n∈NPb0e−β(An2−B|n|)‖P (n)|φ0B〉‖≤ 2∑Pp=Pb+1|Xp| ∑n∈NPb0Pb∏p=1e−β(An2p−Bnp)‖φp‖2npnp!.(4.31)In the exponent, An2p−Bnp can be bounded from below by a linear functionof np of arbitrary slope. That is, for any (fixed) γ > 0,An2p −Bnp ≥ γnp −(B + γ)22A. (4.32)Hence,∣∣∣〈φ0B |TrF[e−βK]|φ0B〉∣∣∣ ≤ 2∑Pp=Pb+1|Xp| ∑n∈NPb0Pb∏p=1e−β(γnp−(B+γ)24A)‖φp‖2npnp!≤ 2∑Pp=Pb+1|Xp|eβPb(B+γ)24APb∏p=1∞∑np=0(e−βγ‖φp‖2)npnp!≤ 2∑Pp=Pb+1|Xp|eβPb(B+γ)24APb∏p=1ee−βγ‖φp‖2≤ 2∑Pp=Pb+1|Xp|eβPb(B+γ)24A e[e−βγ∫dy φ∗0B (y)·φ0B (y)].(4.33)The same bound holds for∣∣∣∣∣〈φ0B |TrF[(e−βqKIR(q))q−1e−βqK]|φ0B〉∣∣∣∣∣.73Hence, the integrand in (4.30) is bounded by2 · 2∑Pp=Pb+1|Xp|eβPb(B+γ)24APb∏p=1χRp(q)(|φp(x)|) e−(1−e−βγ)∫φ∗0B (y)·φ0B (y)dy≤2 · 2∑Pp=Pb+1|Xp|eβPb(B+γ)24A e−(1−e−βγ)∫φ∗0B (y)·φ0B (y)dy,which is independent of q and integrable with respect to the measurePb∏p=1∏x∈Xpdφ∗p(x)dφp(x),since e−βγ < 1. An application of the dominated convergence theorem com-pletes the proof.4.3 The ActionIn this section, we analyze the action,F (,α∗,φ) := log 〈α|e−K |φ〉,which is the logarithm of the inner product that appears in the functionalintegral representation of Theorem 4.2.1.Lemma 4.3.1. For each  > 0, there is a function F (,α∗,φ) that isGrassmann-valued, analytic in the boson fields α∗1, . . . , α∗Pb , φ1, . . . , φPb in aneighbourhood of the origin and even in the Grassmann fields such thateF (,α∗,φ) = 〈α|e−K |φ〉. (4.34)F satisfies the differential equation,∂F∂= −∫X2dxdy α∗(x)k(x,y)∑P(−1)P∏U∈P∂|U |F∂α∗(yU), (4.35)where the sum is over partitions P of Iy. The product over U ∈ P is orderedfrom left to right by decreasing highest element.74The initial condition isF (0,α∗,φ) =∫dx α∗(x) ·φ(x) ≡P∑p=1∫Xpdx α∗p(x)φp(x). (4.36)Proof. Since 〈α|e−K |φ〉 is an entire-analytic function of the boson fieldsα∗1, . . . , α∗Pb and φ1, . . . , φPb , F (,α∗,φ) is also analytic in any region where〈α|e−K |φ〉 is nonzero. When  = 0,〈α|e−K |φ〉 = 〈α|φ〉 = e∫dx α∗(x)·φ(x) 6= 0.By bounding ∂∂〈α|e−K |φ〉, it is easy to explicitly find a small neighbour-hood around the origin in which 〈α|e−K |φ〉 is nonzero.In order to obtain the differential equation (4.35), we differentiate both sidesof the equation eF (,α∗,φ) = 〈α|eK |φ〉 with respect to .eF (,α∗,φ)∂F∂(,α∗,φ) = −〈α|Ke−K |φ〉= −〈α|∫X2dxdy a†(x)k(x,y)a(y) e−K |φ〉= −∫X2dxdy α∗(x)k(x,y)∂∂α∗(y)〈α|e−K |φ〉= −∫X2dxdy α∗(x)k(x,y)∂∂α∗(y)eF (,α∗,φ).We now claim that∂∂α∗(y)eF =∑P(−1)P∏U∈P∂|U |F∂α∗(yU)eF ,where the sum is over partitions P of Iy and the product is ordered bydecreasing largest element in U ∈ P . The claim is trivial in the case |n(y)| =1. Suppose that it also holds for |n(y)| = n − 1 and consider any y ∈Xn11 × · · · × XnPP , with∑p np = n. Without loss of generality, assume thatnP ≥ 1. Let y′ be the projection of y onto Xn11 × · · · ×XnP−1P . (That is, y′75is obtained by removing the last particle coordinate from y.) Then,∂∂α∗(y)eF =∂∂α∗P (y(P,nP ))∂∂α∗(y′)eF=∂∂α∗P (y(P,nP ))∑P ′(−1)P′ ∏U∈P ′∂|U |F∂α∗(y′U)eF=∑P ′(−1)P′ ∂∂α∗P (y(P,nP ))|P ′|∏j=1∂|Uj |F∂α∗(y′Uj)eFwhere the sum is over partitions P ′ of Iy′ . In the third line, we’ve enumeratedthe sets U according to the order of their greatest element. In particular, U1contains the greatest element of all the sets U ∈ P ′, which is (P, nP − 1) ifnP > 1, or (P − 1, nP−1) otherwise. The product is ordered by increasingj, which corresponds to the same order in the line above. We now applythe product rule, taking care to keep track of any minus signs that arisefrom the anticommutation relations which play a role when α∗P (y(P,nP )) is aGrassmann variable. Recall from Equations (4.7) and (4.8),n(f)p (U) := |{m ∈ N : (p,m) ∈ U}| δ[Pb+1,P ](p).Then∂∂α∗(y)eF =∑P ′(−1)P′|P ′|∑k=1(−1)∑k−1j=1n(f)p (Uj)×k−1∏j=1∂|Uj |F∂α∗(yUj)∂|Uk|+1F∂α∗(y{(P,nP )}∪Uk)|P ′|∏j=k+1∂|Uj |F∂α∗(yUj)+ (−1)nP (y′)|P ′|∏j=1∂|Uj |F∂α∗(y′Uj)∂F∂α∗P (y(P,nP )) eF=∑P ′(−1)P′|P ′|+1∑k=1(−1)∑Pp=1∑k−1j=1n(f)p (Uj)n(f)p (Uk)×∂|Uk|+1F∂α∗(y{(P,nP )}∪Uk)k−1∏j=0∂|Uj |F∂α∗(yUj)|P ′|∏j=k+1∂|Uj |F∂α∗(yUj) eF=∑P(−1)P∏U∈P∂|U |F∂α∗(yU)eF ,76where U|P ′|+1 (in the second line above) is the empty set. In the last line, thesum is over partitions P of Iy. Each P is equal to{U1, . . . , Uk ∪ {(P, nP )}, . . . , U|P ′|}for some partition P ′ = {U1, · · · , U|P ′|} of Iy′ and some k. The sign ofsuch P , (−1)P , is precisely equal to (−1)∑Pp=1∑k−1j=1n(f)p (Uj)n(f)p (Uk)(−1)P′. Thiscompletes the proof.Lemma 4.3.2. The differential equation (4.35) satisfied by F can also beexpressed as∂F∂= −∑`≥1∑m1,...,m` 6=0mj∈NP0(∑`j=1 mj)!`!∫Xdx∏`j=1(∫Xmjdyj)α∗(x)k(x,y1 ◦ · · · ◦ y`)∏`j=11mj!∂|mj |F∂α∗(yj),where for any m = (m1, . . . ,mP ) ∈ NP0 ,m! :=P∏p=1mp!.Recall also that y1 ◦ · · · ◦y` is the vector formed by concatenating y1, . . . ,y`,as defined in Equations (4.4) and (4.5).Proof. Recall that each partition P of Iy is a set of disjoint subsets U ofIy such that unionsqU∈PU = Iy. If we arbitrarily order the elements of P asU1, U2, · · · , U|P|, we will denote the vector (U1, U2, · · · , U|P|) by ~P and call itan ordered partition. For a given partition P , there are |P|! correspondingordered partitions ~P . We will also let | ~P| := |P|. The sign (−1)~P of theordered partition ~P is the sign of the permutation required to bring the co-ordinates of each fermionic species from the concatenated list ~U1 ◦ · · · ◦ ~U|P|back to their order in Ix, where the elements of each ~Uj is arranged in in-creasing order.If we sum over ordered partitions in Equation (4.35) rather than partitions,77we obtain,∂F∂=−∫X2dxdy α∗(x)k(x,y)|n(y)|∑`=11`!∑~P|~P|=`(−1)~P∏`j=1∂|Uj |F∂α∗(yUj).Since the kernel is symmetric in the boson coordinates and antisymmetricin the fermion coordinates, rearranging the arguments of y to the orderyU1 ◦ · · · ◦ yU` results in a sign of (−1)~P . Hence,∂F∂=−∑`≥11`!∑|~P|=`∫X2dxdy α∗(x)k(x,yU1 ◦ · · · ◦ yU`)∏`j=1∂|Uj |F∂α∗(yUj)=−∑`≥11`!∑m1,...,m` 6=0(m1 + · · ·+ m`)!m1! · · ·m`!∫Xdx∫Xm1dy1 · · ·∫Xm`dy`α∗(x)k(x,y1 ◦ · · · ◦ y`)∏`j=1∂|mj |F∂α∗(yj).In the second line, we essentially replaced each set Uj with its correspondingvector mj := n(Uj) := (n1(Uj), . . . , nP (Uj)) of particle numbers. Each yjis an independent variable with mj coordinates. The number of orderedpartitions corresponding to each sequence {mj} is equal to the multinomialcoefficient (m1+···+m`)!m1!···m`! . This completes the proof.In the lemma below, for each p ∈ {1, . . . , P}, let ep be the unit vector,ep := (0, . . . , 0, 1, 0, . . . , 0) ∈ NP0 , (4.37)where the 1 lies in the pth argument.Theorem 4.3.3. F has an expansionF (,α∗,φ) =∑n∈NP0∫Xndx∫Xndy α∗(x)Fn(,x,y)φ(y), (4.38)where the coefficients Fn(,x,y) ∈ C are given by the following recursion78relations. For each p ∈ {1, . . . , P},Fep(, x, y) = (e−hp)(x, y), (4.39)where hp = P (ep)KP (ep) is the single-particle operator for species p (whichwould include both the kinetic energy and the chemical potential). In otherwords, the kernel of hp satisfieshp(x, y) = k(xep, yep).For all n with |n| > 1 and x,x′ ∈ Xn, letHn(x,x′) := S(n)x∑06=m≤nn!(n−m)!k(x[≤m],x′[≤m])δx[>m](x′[>m]). (4.40)Given ` ∈ N, let n := (n1, . . . ,n`) where each nj ∈ NP0 and∑`j=1 nj = n.Then for all x ∈ Xn and x′j ∈ Xnj where j ∈ {1, . . . , `}, letKn(x,x′1, . . . ,x′`) =S(n)x∑(mj)`j=106=mj≤njm!`!n!m!(n−m)!(−1)P∑p=Pb+1∑`j=2(njp−mjp )j−1∑k=1nkpk(x[≤m],x′1[≤m1] ◦ · · · ◦ x′`[≤m`])δx˜1(x′1[>m1]) . . . δx˜`(x′`[>m`])(4.41)where m :=∑`j=1 mj, n! :=∏`j=1 nj! and m! :=∏`j=1 mj!. Each x˜j ∈ Xnj−mjis defined by x[>m] = x˜1 ◦ · · · ◦ x˜`.Then, for all n with |n| > 1,Fn(,x,y) = −∫ 0dτ∫Xndx′ e−(−τ)Hn(x,x′)∑`>1∑n=(nj)`j=1∑nj=nnj 6=0∫Xn`dx′′` · · ·∫Xn1dx′′1 Kn(x′,x′′1 , . . . ,x′′` )∏`j=1Fnj(τ,x′′j ,yj),(4.42)79where each yj ∈ Xnj is defined byy = y1 ◦ y2 ◦ · · · ◦ y`.Proof. Since F (,α∗,φ) is analytic in α∗, we can make the expansion,F (,α∗,φ) =∑n∈NP0∫Xndx α∗(x)Fn(,x,φ),so thatFn(,x,φ) =1n!∂∂α∗(x)F (,α∗,φ)∣∣∣∣∣α∗=0. (4.43)For a Grassmann variable α∗p with p > Pb, the evaluation at α∗p = 0 is equalto the term of zeroth order in α∗p.We now find a differential equation for each coefficient Fn by differentiatingthe above equation (4.43) with respect to  and applying Lemma 4.3.2 onthe right-hand side. Throughout the calculation below, let m :=∑`j=1 mj.∂Fn∂(,x,φ) = −1n!∂∂α∗(x)∑`≥1∑m1,...,m`6=0m!`!∫Xdz∏`j=1∫Xmjdyjα∗(z)k(z,y1 ◦ · · · ◦ y`)∏`j=11mj!∂|mj |F∂α∗(yj)∣∣∣∣∣∣α∗=0=−1n!S(n)x∑`≥1∑m1,...,m`6=0m!`!∏`j=1∫Xmjdyjn!m!(n−m)!m!k(x[≤m],y1 ◦ · · · ◦ y`)∂∂α∗(x[>m])∏`j=11mj!∂|mj |F∂α∗(yj)∣∣∣∣∣∣α∗=0=− S(n)x∑`≥1∑m1,...,m`6=0m!`!(n−m)!∏`j=1∫Xmjdyj k(x[≤m],y1 ◦ · · · ◦ y`)×∑w1,...,w`≥0∑wj=n−m(n−m)!∏`j=1(−1)γjwj!mj!∂|mj |+|wj |F∂α∗(x˜j)∂α∗(yj)∣∣∣∣∣∣∣∣∣α∗=0,80wherex[>m] = x˜1 ◦ · · · ◦ x˜` with each x˜j ∈ Xwj ,andγj :=P∑p=Pb+1np(x˜j)j−1∑k=1(np(x˜k) + np(yk))=P∑p=Pb+1wjpj−1∑k=1(wkp +mkp).Here, (−1)γj is a sign that arises from the product rule for Grassmann deriva-tives. Noting that∂|mj |+|wj |F∂α∗(x˜j)∂α∗(yj)=∂|mj |+|wj |F∂α∗(yj ◦ x˜j),we can evaluate at α∗ = 0 and use the definition of Fn to conclude that∂Fn∂(,x,φ) = −S(n)x∑`≥1∑m1,...,m`6=0m!`!∏`j=1∫Xmjdyj k(x[≤m],y1 ◦ · · · ◦ y`)∑w1,...,w`≥0∑wj=n−m∏`j=1(mj + wj)!wj!mj!(−1)γjFmj+wj(,yj ◦ x˜j,φ).For each ` ∈ N and j ∈ {1, . . . , `}, setnj = mj + wjandKn(x,x′1, . . . ,x′`) = S(n)x∑(mj)`j=106=mj≤njm!`!n!m!(n−m)!(−1)P∑p=Pb+1∑`j=1(njp−mjp )j−1∑k=1nkpk(x[≤m],x′1[≤m1] ◦ · · · ◦ x′`[≤m`])δx˜1(x′1[>m1]) . . . δx˜`(x′`[>m`]).81where n! :=∏`j=1 nj!. Then,∂Fn∂(,x,φ) = −∑`≥1∑n=(nj)`j=1∑nj=nnj 6=0∫Xn`dx′` · · ·∫Xn1dx′1 Kn(x,x′1, . . . ,x′`)×∏`j=1Fnj(,x′j,φ).Singling out the term involving Fn,∂Fn∂(,x,φ) =−∫Xndx′Hn(x,x′)Fn(,x′,φ)−∑`>1∑n=(nj)`j=1∑nj=nnj 6=0∫Xn`dx′` · · ·∫Xn1dx′1 Kn(x,x′1, . . . ,x′`)∏`j=1Fnj(,x′j,φ),where Hn is equal to Kn in the case where n is a sequence with only oneelement, n = (n). For x,x′ ∈ Xn,Hn(x,x′) = Sx∑06=m≤nn!(n−m)!k(x[≤m],x′[≤m])δx[>m](x′[>m]).In the case n = ep, the differential equation can be simplified to∂Fep∂(, x,φ) =−∫Xpdx′ hp(x, x′)Fn(x′)where hp(x, x′) = k(xep, x′ep).The initial conditions areFn(0,x,φ) =φ(x) if n = ep for some p, and0 otherwise.Using the integrating factor eHn , we solve the differential equation for Fn to82obtain the recursive integral formula,Fn(,x,φ) = −∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑`>1∑n=(nj)`1∑nj=nnj 6=0∫Xn`dx′′` · · ·∫Xn1dx′′1Kn(x′,x′′1 , . . . ,x′′` )∏`j=1Fnj(τ,x′′j ,φ), (4.44)for all |n(x)| > 1. For n = ep,Fep(, x, φ) = (e−hpφ)(x).By induction, Fn(,x,φ) must also be of degree n in φ. LetFn(,x,φ) =∫Xndy Fn(,x,y)φ(y). (4.45)Then∏`j=1Fnj(τ,x′′j ,φ) =∏`j=1∫Xnjdyj Fnj(τ,x′′j ,yj)φ(yj).The recursion relations for Fn(,x,y) becomeFep(, x, y) = (e−hp)(x, y)and, for |n| > 1,Fn(,x,y) = −∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑`>1∑n=(nj)`j=1∑nj=nnj 6=0∫Xn`dx′′` · · ·∫Xn1dx′′1 Kn(x′,x′′1 , . . . ,x′′` )∏`j=1Fnj(τ,x′′j ,yj).where, for each `, y = y1 ◦ · · · ◦ y` with each yj ∈ Xnj . This can be seen byinserting (4.45) into the recursion relation (4.44) and letting y := y1◦· · ·◦y`.83That is,Fn(,x,φ) = −∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑`>1∑n=(nj)`1∑nj=nnj 6=0∫Xn`dx′′` · · ·∫Xn1dx′′1Kn(x′,x′′1 , . . . ,x′′` )∏`j=1∫Xnjdyj Fnj(τ,x′′j ,yj)φ(yj)=∫Xdy−∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑`>1∑n=(nj)`j=1∑nj=nnj 6=0∫Xn`dx′′` · · ·∫Xn1dx′′1 Kn(x′,x′′1 , . . . ,x′′` )∏`j=1Fnj(τ,x′′j ,yj)φ(y).This completes the proof.844.3.1 Bounds on Fn for 2-Body InteractionsWe now specialize to the case where K has order M = 2. In particular,K =P∑p=1∫X2pdx dy ap(x)†hp(x, y)ap(y)+P∑p=1P∑q=1∫Xpdx∫Xqdy a†p(x)a†q(y)vpq(x, y)ap(x)aq(y), (4.46)where for all p, q ∈ {1, . . . , P}, hp : X2p → R, vpq : Xp ×Xq → R, and for allx ∈ Xp and y ∈ Xq,vpq(x, y) = vqp(y, x).In the notation used thus far in this chapter whereK =∫X2dx dy a†(x)k(x,y)a(y),the kernel k of K in this case satisfiesk(x,y) =hp(x, y) if x = (x) ep and y = (y) ep,vpq(xp, xq)δxp(yp)δxq(yq) if p 6= q, x = (xp)ep + (xq)eqand y = (yp)ep + (yq)eq,vpp(xp1, xp2)δxp1(yp1)δxp2(yp2) if x = (xp1, xp2)ep andy = (yp1, yp2)ep,0 otherwise.We assume that K is repulsive in the sense thatλ0(v) := infP∑p,q=1∫Xpdx∫Xqdy ρp(x)vpq(x, y)ρq(y)∣∣∣∣∣∣P∑p=1∫Xpdx ρp(x)2 = 1, (∀p)(∀x ∈ Xp)ρq(x) ≥ 0(4.47)is (strictly) positive for all p ∈ {1, . . . , P}.85The function Hn(x,x′) defined in Equation (4.40) asHn(x,x′) := S(n)x∑0 6=m≤nn!(n−m)!k(x[≤m],x′[≤m])δx[>m](x′[>m])may be simplified in this case toHn(x,x′) =P∑p=1np∑j=1hp(xpj , x′pj)∏(q,i)6=(p,j)δxqi (x′qi)+P∑p=1np∑i=1P∑q=1nq∑j=1vpq(xpi , xqj)− vpp(xpi , xpi) δx(x′).We definehn(x,x′) :=P∑p=1np∑j=1hp(xpj , x′pj)∏(q,m)6=(p,j)δxqm (x′qm)andVn(x,x′) :=P∑p=1np∑i=1P∑q=1nq∑j=1vpq(xpi , xqj)− vpp(xpi , xpi) δx(x′)so that Hn(x,x′) = hn(x,x′) + Vn(x,x′).Recall from Equation (4.41), for ` > 1,Kn(x,x′1, . . . ,x′`) =S(n)x∑(mj)`j=1(∀j)0 6=mj≤njm!`!n!m!(n−m)!(−1)P∑p=Pb+1∑`j=1(njp−mjp )j−1∑k=1nkpk(x[≤m],x′1[≤m1] ◦ · · · ◦ x′`[≤m`])δx˜1(x′1[>m1]) . . . δx˜`(x′`[>m`]),where each x˜j ∈ Xnj−mj is defined by x[>m] = x˜1 ◦ · · · ◦ x˜`.Since each mj 6= 0 and k(x[≤m],x′1[≤m1] ◦ · · · ◦x′`[≤m`]) = 0 unless |m| ≤ 2, wemust conclude that the only nonzero terms in the above equation have ` = 286and |m1| = |m2| = 1.In this case, we can simplify Kn to the form,K(n1,n2)(x,x′1,x′2) = S(n)xP∑p=1P∑q=pn1pn2q(−1)sq(n)vpq(xp1 , xqn1q+1)δx(x′1 ◦ x′2).(4.48)where, for each p ∈ {1, . . . , P},sp(n) :=∑p 6=p′>Pbn2p′n1p′ + χ[>Pb](p)(n2p − 1)n1pandχ[>Pb](p) =1 if p > Pb0 otherwise.With ` = 2, the recursion relation (4.42) from Lemma 4.3.3 becomesFn(,x,y) =−∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑n1,n2 6=0n1+n2=n∫Xn2dx′′2∫Xn1dx′′1K(n1,n2)(x′,x′′1 ,x′′2)Fn1(τ,x′′1 ,y[≤n1])Fn2(τ,x′′2 ,y[>n1]). (4.49)By substituting K(n1,n2)(x′,x′′1 ,x′′2) with the expression from Equation (4.48),Fn(,x,y) =−∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑n1,n2 6=0n1+n2=n∫Xn2dx′′2∫Xn1dx′′1S(n)x′P∑p=1P∑q=pn1pn2q(−1)sq(n)vpq(x′p1 , x′qn1q+1)δx′(x′′1 ◦ x′′2)Fn1(τ,x′′1 ,y[≤n1])Fn2(τ,x′′2 ,y[>n1])=−∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑n1,n2 6=0n1+n2=nS(n)x′P∑p=1P∑q=pn1pn2q(−1)sq(n)vpq(x′p1 , x′qn1q+1)·87· Fn1(τ,x′[≤n1],y[≤n1])Fn2(τ,x′[>n1],y[>n1])=−∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑n1,n2 6=0n1+n2=nS(n)x′P∑p=1P∑q=pn1p∑i=1np∑j=n1q+1(−1)sq(n)vpq(x′pi , x′qj)Fn1(τ,x′[≤n1],y[≤n1])Fn2(τ,x′[>n1],y[>n1]).Definition 13. For any n ∈ NP0 and F : Xn → C, we define‖F‖1,∞ = maxp∈{1,...,P}max1≤j≤npmaxxj∈Xp∫Xn−ep∏(q,i) 6=(p,j)dxqi |F (x)|.Lemma 4.3.4. For any n ∈ NP0 and F : Xn ×Xn → C,(a) ‖e−hnF‖1,∞ ≤ e|n|h0‖F‖1,∞, where h0 := maxp ‖hp‖1,∞,(b) ‖e−VnF‖1,∞ ≤ e|n|v0‖F‖1,∞, where v0 := maxp,x∈Xpvpp(x, x), and(c) ‖e−HnF‖1,∞ ≤ e(h0+v0)|n|‖F‖1,∞.Proof. The proof is very similar to that of Lemma (3.10) in [10].(a) For each p, leth(p)n (x,x′) :=np∑j=1hp(xpj, x′pj)∏(q,m)6=(p,j)δxqm(x′qm),so that hn =∑Pp=1 h(p)n . From [10], we know that for each p,‖e−hp‖1,∞ ≤ e‖hp‖1,∞and‖e−h(p)n F‖1,∞ ≤ enp‖hp‖1,∞‖F‖1,∞.(The proof is the same, regardless of whether the pth species is bosonic88or fermionic.) Since h(p)n and h(q)n commute,‖e−hnF‖1,∞ =∥∥∥∥∥∥P∏p=1e−h(p)n F∥∥∥∥∥∥1,∞≤P∏p=1enp‖hp‖1,∞‖F‖1,∞= e|n|h0‖F‖1,∞,where h0 := max1≤p≤P‖hp‖1,∞.(b) Recall,Vn(x,x′) :=P∑p=1np∑i=1P∑q=1nq∑j=1vpq(xpi , xqj)− vpp(xpi , xpi) δx(x′).Since V is repulsive, we can let ρp(x) = 1√Pnp∑npi=1 δxpi (x) in Equation(4.47) to show thatP∑p,q=1np∑i=1nq∑j=1vpq(xpi , xqj) ≥ λ0(v) > 0.Hence,P∑p=1np∑i=1P∑q=1nq∑j=1vpq(xpi , xqj)− vpp(xpi , xpi) ≥ −P∑p=1np∑i=1vpp(xpi , xpi)≥ −|n|v0,wherev0 := max1≤p≤Pmaxx∈Xpvpp(x, x).Therefore,|e−VnF | ≤ e|n|v0|F |.The claim follows.(c) Part (c) follows from the Trotter product formula and Parts (a) and(b), as in [10].89Theorem 4.3.5. For a system with P species of bosons and/or fermionsinteracting with a repulsive 2-body potential as defined in Equations (4.46)and (4.47), the coefficients Fn(,x,y) in the expansion (4.38) for the actionF (,x,y) satisfy the bound‖Fn(, ·, ·)‖1,∞ ≤ (λv)|n|−1e(h0+v0)|n|1|n|P+2,where |n| :=∑p np andλv := 2P+2P∑p,q=1‖vp,q‖1,∞∑06=m∈NP01|m|P+1. (4.50)Proof. For convenience, within this proof, let C1 := h0 +v0. In the case |n| =1, n = ep for some p, and from Equation (4.39), Fep(, x, y) = (e−hp)(x, y).Hence,‖Fn(, ·, ·)‖1,∞ =‖e−hp‖1,∞ ≤ e‖hp‖1,∞ .For any n ≥ 2, assume the bound for all Fm with |m| =∑pmp < n. Thenfor any n with |n| = n,‖Fn(,x,y)‖1,∞ =∥∥∥∥∥∥∥∥∫ 0dτ∫Xndx′e−(−τ)Hn(x,x′)∑n1,n2 6=0n1+n2=nS(n)x′P∑p=1P∑q=pn1p∑i=1np∑j=n1q+1(−1)sq(n)vpq(x′pi , x′qj)Fn1(τ,x′[≤n1],y1)Fn2(τ,x′[>n1],y2)∥∥∥∥∥∥∥∥1,∞≤∫ 0dτ∑n1,n2 6=0n1+n2=nP∑p=1P∑q=pn1p∑i=1np∑j=n1q+1∥∥∥∥∫Xndx′e−(−τ)Hn(x,x′)S(n)x′vpq(x′pi , x′qj)Fn1(τ,x′[≤n1],y1)Fn2(τ,x′[>n1],y2)∥∥∥∥1,∞≤∫ 0dτ∑n1,n2 6=0n1+n2=n∑p,qp≤qn1pn2qe(h0+v0)|n|(−τ)‖vpq‖1,∞90· ‖Fn1(τ, ·, ·)‖1,∞‖Fn2(τ, ·, ·)‖1,∞≤∫ 0dτ∑n1,n2 6=0n1+n2=n∑p,qp≤qn1pn2qeC1|n|(−τ)‖vpq‖1,∞(λvτ)|n1|−1eC1|n1|τ1|n1|P+2(λvτ)|n2|−1eλv |n1|τ1|n2|P+2≤∫ 0dτ∑n1,n2 6=0n1+n2=n∑p,qp≤qn1pn2qeC1|n|(−τ)‖vpq‖1,∞(λvτ)n−2eC1nτ1|n1|P+2|n2|P+2≤ eC1|n|(∫ 0dτ(λvτ)n−2)∑n1,n2 6=0n1+n2=n∑p,qp≤qn1pn2q‖vpq‖1,∞1|n1|P+2|n2|P+2≤ eC1|n|(λv)n−1n− 11λv∑n1,n2 6=0n1+n2=n∑p,qp≤qn1pn2q‖vpq‖1,∞1|n1|P+2|n2|P+2≤ eC1|n|(λv)n−11λv∑p,qp≤q‖vpq‖1,∞1n− 1∑m≤n06=m 6=n1|m|P+1|n−m|P+1.The sum over m has a symmetry about m = 12n. Hence,∑m≤n06=m6=n1|m|P+1|n−m|P+1≤ 2∑m≤ 12n06=m 6=n1|m|P+1|n−m|P+1≤ 21(12 |n|)P+1∑m≤ 12n06=m6=n1|m|P+1≤2P+2nP+1∑m∈NP0m6=01|m|P+1.This is finite, since∑m∈NP0m 6=01|m|P+1≤ C∫RP‖x‖>1dx1‖x‖P+1≤ C ′∫ ∞11rP+1rP−1dr91= C ′∫ ∞11r2dr <∞,for some constants C and C ′. Hence, if we letλv := 2P+2P∑p,q=1‖vpq‖1,∞∑m∈NP0m 6=01|m|P+1,then‖Fn(,x,y)‖1,∞ ≤ (λv)|n|−1eC1|n|1|n|P+2,as claimed. This completes the proof.Proposition 4.3.6. For each  > 0, the action, F (,α∗,φ) is an analyticfunction of the fields α∗ and φ on the domain defined by|α∗p|Xp , |φp|Xp <e−12 (h0+v0)√λv,for all p ≤ Pb, where | · |Xp is the supremum norm on Xp,|φp|Xp := maxx∈Xp|φp(x)|.As defined in Theorem 4.3.5, λv is the constant,λv = 2P+2P∑p,q=1‖vp,q‖1,∞∑06=m∈NP01|m|P+1.Proof. For each n = (n1, . . . , nP ) ∈ NP0 , let nb represent the vector of onlythe boson numbers,nb := (n1, . . . , nPb)and let nf represent the vector of only the fermion numbers,nf := (nPb+1, . . . , nP ).92Then n = nb ◦ nf . Similarly, for each x = (~x1, . . . , ~xP ) ∈ Xn, letxb := (~x1, . . . , ~xPb) ∈ Xnbandxf := (~xPb+1, . . . , ~xP ) ∈ Xnf ,so that x = xb ◦ xf .With this notation,F (,α∗,φ) =∑n∈NP0∫Xndx∫Xndy α∗(x)φ(y)Fn(,x,y)=∑nf∈{1,...,|Xf |}Pf∫Xnfdxf∫Xnfdyf α∗(xf )φ(yf )·∑nb∈NPb0∫Xnbdxb∫Xnbdyb α∗(xb)φ(yb)Fn(,x,y)Since the outer sum over fermion numbers is finite, we can focus on the innersum over boson numbers. Using the bound from Theorem 4.3.5,∑nb∈NPb0∫Xnbdxb∫Xnbdyb |α∗(xb)φ(yb)Fn(,x,y)|≤∑nb∈NPb0Pb∏p=1|αp|Xp |φp|Xp∫Xnbdxb∫Xnbdyb |Fn(,x,y)|≤(maxp|Xp|)∑nb∈NPb0Pb∏p=1|αp|npXp|φp|npXp ‖Fn(, ·, ·)‖1,∞≤(maxp|Xp|)∑nb∈NPb0Pb∏p=1(|αp|Xp|φp|Xp)np(λv)|n|−1e(h0+v0)|n|1|n|P+2≤(maxp|Xp|)(λv)|nf |−1e(h0+v0)|nf |Pb∏p=1∞∑np=0(|αp|Xp|φp|Xpe(h0+v0)λv)np.93This is absolutely convergent as long as, for each p ≤ Pb,|αp|Xp |φp|Xp <1e(h0+v0)λv,which is certainly true in the domain with|αp|Xp , |φp|Xp <e−12 (h0+v0)√λv.This completes the proof.Note that it is easy to choose the large-field cutoffs Rp(q) in Theorem 4.2.1so that they satisfy both the analyticity conditionRp(q) <e−12 (h0+v0)βq√λvβq(4.51)and the convergence conditionlimq→∞qe−12Rp(q)2= 0. (4.52)For example,Rp(q) = C√q, with C =1√λvβe−12 (h0+v0)β,clearly satisfies both conditions (4.51) and (4.52) above.Hence, we can replace each inner product 〈φτ−|e−K |φτ 〉 in the functionalintegral (4.26) with an exponential of the action F (,φ∗τ−,φτ ) to obtainTr e−βK = limq→∞∫ ∏τ∈Tq[dµR(q)(φ∗τ ,φτ )e−∫dy φ∗τ (y)·φτ (y)] ∏τ∈TqeF (,φ∗τ−,φτ ).We have assumed here that the interactions are of order at most M = 2and that, for each p ∈ {1, . . . , P}, Rp(q) satisfies both conditions (4.51) and(4.52) above.94Chapter 5Systems of Electrons andPhononsWe now consider a system of bosons and fermions, with a particular Hamilto-nian, known as the Holstein Hamiltonian, that does not preserve the numberof bosons. If a†b(x), ab(x) are the creation and annihilation operators at apoint x of the boson configuration space Xb and a†f (x), af (x) are the creationand annihilation operators at a point x of the fermion configuration spaceXf , the operator representing the interaction between bosons and fermionshas the formV =∫Xbdx∫Xfdy v(x, y)a†f (y)af (y)(a†b(x) + ab(x)),where v : Xb ×Xf → R. This type of interaction is often used for modellingsystems of electrons and phonons [14, 15, 20].In Section 5.1, we’ll provide a brief review of the notation defined in Chap-ters 2-4, as applied to a system of one species of bosons and one speciesof fermions. In Section 5.2, we’ll define the Hamiltonian and prove boundson it which will be used repeatedly throughout the remainder of the Chap-ter. In Section 5.3, we’ll derive the functional integral representation, whichtakes the same form as the functional integral in (4.26) for particle-preservinginteractions. However, the non-preservation of boson number will demandnew techniques in the proof. In particular, we’ll use a Duhamel expansionin powers of V . Finally, in Section 5.4, we’ll show that the “action” is anentire-analytic non-zero function of the fields.955.1 NotationWe denote the configuration space for the bosons by Xb and the configura-tion space for fermions by Xf . Each of these incorporate the position spaceas well as any internal states, such as spin.The Hilbert space for a system of indistinguishable bosons on the configura-tion space Xb isB(Xb) =∞⊕n=0Bn(Xb)where Bn(Xb) is the space of symmetric functions of n arguments on Xb,Bn(Xb) := {f ∈ L2(Xnb ) : (∀σ ∈ Sn)σf = f}.Similarly, the Hilbert space for a system of indistinguishable fermions on theconfiguration space Xf isF(Xf ) =|Xf |⊕n=0Fn(Xf ),whereFn(Xf ) := {f ∈ L2(Xnf ) : (∀σ ∈ Sn)σf = (−1)σf}.The Hilbert space for the combined system of bosons and fermions isH := B(Xb)⊗F(Xf ).Throughout this chapter, we will sometimes suppress the configuration spacesin the notation above and denote B(Xb) and F(Xf ) by B and F respectively.Similarly, the n-boson and n-fermion subspaces of B and F will be denotedby Bn and Fn respectively.We denote the n-boson subspace of H byHn := Bn ⊗F ,while the projection onto H(n) is denoted by P (n). The subspace of H with96at most n bosons isH[≤n] :=n∑m=1Hm,and Pn is the projection onto this space.Recall from Section 2.1.1 that for each n ∈ N0, {δY : Y ∈ Xnb /Sn} is a basisfor Bn(Xb) and {δY : Y ⊂ Xf} is a basis for F(Xf ).The power set of Xf is the set of all subsets of Xf and is denoted by P(Xf ).We also denote the set of all subsets with exactly n elements by Pn(Xf ),Pn(Xf ) := {Y ⊂ Xf : |Y | = n}.For any Y = (Yb, Yf ) ∈ Xnb/Snb × P(Xf ), we let δY := δYb ⊗ δYf . Then{δY : Y ∈ Xnb/Snb × P(Xf )}is an orthonormal basis for Hnb . The setB(H) :={δY : Y ∈(∞⊔n=0Xnb /Sn)× P(Xf )}is an orthonormal basis for H.For any n = (nb, nf ) ∈ N20, letXn := Xnbb ×Xnff .Then letX :=⊔n∈N20Xn.As with all other finite sets in this work, integrals over X and any subsetthereof will be with respect to the counting measure. For example, if f :X→ C, ∫Xdx f(x) =∑x∈Xf(x).For each x ∈ X, we let n(x) be the element n = (nb, nf ) of N20 such that97x ∈ Xn. Then there is a vector ~xb of boson coordinates,~xb = (xb1, . . . , xbnb),and a vector of fermion coordinates,~xf = (xf1, . . . , xfnf ),such that x = (~xf , ~xb). In this case, the creation and annihilation operatorsat x area†(x) := a†b(xb1) . . . a†b(xbnb)a†f (xf1) . . . a†f (xfnf )anda(x) := af (xfnf ) . . . af (xf1)ab(xbnb) . . . ab(xb1).respectively.5.2 The HamiltonianThe Hamiltonian H will be defined as the sum of two operators, H0 and V .The kinetic energy operator H0 is defined byH0 :=∫X2bdx dy a†b(x)hb(x, y)ab(y) +∫X2fdx dy a†f (x)hf (x, y)af (y),where hb is a complex-valued function on X2b that satisfies hb(x, y) = h∗b(y, x)for all x, y ∈ Xb. Likewise, hf : X2f → C with hf (x, y) = h∗f (y, x) for allx, y ∈ Xf . H0 is well-defined as a map from the finite-dimensional space Hnto Hn, for each n ∈ N, and may be extended linearly to the domain,H(finite) = {ψ ∈ H : (∃N ∈ N)(∀n ≥ N)P (n)ψ = 0}.Lemma 5.2.1. The kinetic energy operator H0 is essentially self-adjoint onH(finite).Proof. Due to the symmetry conditions on the functions hb and hf , H0 issymmetric. As a symmetric operator on a finite-dimensional space, eachH0|Hn satisfies Ran(H0|Hn ± i) = Hn. Hence, with domain H(finite), the98operator H0 ± i has range H(finite) as well, which is dense in H. It followsthat H0 is essentially self-adjoint on H(finite).Note that H0 satisfies the bounds,λ ·N ≤ H0 ≤ Λ ·N, (5.1)where N := (Nb, Nf ) the vector of particle number operators. On the leftside, λ = (λb, λf ) and λb, λf are the lowest eigenvalues of hb and hf re-spectively. On the right side, Λ = (Λb,Λf ), where Λb, Λf are the largesteigenvalues of hb and hf respectively. We assume that the lowest eigenvalueof hb, λb is strictly positive.The energy due to interaction between the bosons and fermions is representedby the operator,V :=∫Xbdx∫Xfdy v(x, y)a†f (y)af (y)(ab(x) + a†b(x)),where v is a real-valued function on Xb ×Xf . This operator is well-definedoperator on each Hn. Although it preserves fermion number, V does notpreserve boson number. When applied to an n-boson state, this operatorwill return a sum of (n− 1)-boson and (n+ 1)-boson states. Hence, V mapseach Hn to Hn−1 ⊕ Hn+1, and may be extended linearly to H(finite). It iseasy to see that V is a symmetric operator on H(finite).Lemma 5.2.2. The interaction operator V satisfies the following bounds.(a) For all m,n ∈ N0,‖P (n)V P (m)‖ ≤12Cv√n+ 1 if m = n+ 1,12Cv√n if m = n− 1,0 otherwise.(5.2)whereCv := 2|Xb||Xf |‖v‖∞. (5.3)99and‖v‖∞ := maxx,y∈Xb×Xf|v(x, y)|.(b) On the domain H(finite),− Cv√Nb + 1 ≤ V ≤ Cv√Nb + 1. (5.4)Proof. (a) Recall from Proposition 2.2.4 that the boson creation and anni-hilation operators satisfy the bounds,‖ab(x)P (n)‖ ≤√n and ‖a†b(x)P(n)‖ ≤√n+ 1,for any x ∈ Xb. Also, the fermion number operator at y ∈ Xf satisfies‖nf (y)‖ = ‖a†f (y)af (y)‖ ≤ 1. Hence,‖P (m)V P (n)‖=∥∥∥∥∥P (m)∫Xbdx∫Xfdy v(x, y)a†f (y)af (y)(ab(x) + a†b(x))P (n)∥∥∥∥∥≤ ‖v‖∞|Xf |∫Xbdx∥∥∥P (m)(ab(x) + a†b(x))P (n)∥∥∥≤‖v‖∞|Xf ||Xb|√n+ 1 if m = n+ 1,‖v‖∞|Xf ||Xb|√n if m = n− 1,0 otherwise.(b) Let ψ be an arbitrary element ofH(finite). Denote the n-boson projection,P (n)ψ, by ψn for all n ∈ N. For convenience, let φ−1 := 0. Sinceψ ∈ H(finite), we can choose an integer N such that ψn = 0 for alln > N . From Part a,〈ψ, V ψ〉 =∞∑n=0(〈ψn, V, ψn+1〉+ 〈ψn, V, ψn−1〉)≤ |Xb||Xf |‖v‖∞∞∑n=0(√n+ 1 ‖ψn‖‖ψn+1‖+√n ‖ψn‖‖ψn−1‖)100= 2|Xb||Xf |‖v‖∞∞∑n=0√n+ 1 ‖ψn‖‖ψn+1‖≤ |Xb||Xf |‖v‖∞∞∑n=0√n+ 1(‖ψn‖2 + ‖ψn+1‖2)≤ 2|Xb||Xf |‖v‖∞∞∑n=0√n+ 1 ‖ψn‖2= 2|Xb||Xf |‖v‖∞〈ψ,√Nb + 1 ψ〉.By the same reasoning,〈ψ, V ψ〉 ≥ −2|Xb||Xf |‖v‖∞〈ψ,√Nb + 1 ψ〉.Hence,−Cv√Nb + 1 ≤ V ≤ Cv√Nb + 1,as claimed.Proposition 5.2.3. The Hamiltonian,H := H0 + V=∫X2bdx dy a†b(x)hb(x, y)ab(y) +∫X2fdx dy a†f (x)hf (x, y)af (y)+∫Xbdx∫Xfdy v(x, y)a†f (y)af (y)(ab(x) + a†b(x))is essentially self-adjoint on the domain H(finite).Proof. From Lemma 5.2.1, H0 is essentially self-adjoint on H(finite). By theKato-Rellich Theorem, Theorem X.12 of [1], if we can find real constants aand b, with a < 1, such that‖V φ‖ ≤ a‖H0φ‖+ b‖φ‖, (5.5)for all φ ∈ H(finite), the proof will be complete.Let φ be an arbitrary element of H(finite) and let φn := P (n)φ for all n ∈ N.For convenience, let φ−1 := 0. Since φ ∈ H(finite), we may choose N ∈ N101such that φn = 0 for all n > N . Then, φ =∑Nn=0 φn and‖V φ‖2 =N+1∑n=0‖P (n)V φ‖2=N+1∑n=0‖P (n)V (φn−1 + φn+1)‖2≤ (‖v‖∞|Xf ||Xb|)2N+1∑n=0(√n‖φn−1‖+√n+ 1‖φn+1‖)2≤12C2vN∑n=0(2n+ 1)‖φn‖2,where Cv = 2|Xb||Xf |‖v‖∞, as defined in Lemma 5.2.2. Noting that n ≤n2M +M for any M > 0, let M =C2v2λ2bfor any  > 0, where λb is the smallesteigenvalue of hb. Then,‖V φ‖2 ≤N∑n=0(2λ2bn2 +C4v2λ2b+C2v2)‖φn‖2≤ 2λ2b〈φ,N2b φ〉+(C4v2λ2b+C2v2)‖φ‖2≤ 2〈φ,H20φ〉+(C4v2λ2b+C2v2)‖φ‖2= 2‖H0φ‖2 +(C4v2λ2b+C2v2)‖φ‖2.With a =  and b =√C4v2λ2b+ C2v2 , the inequality (5.5) holds. This completesthe proof.Since our goal is to find a functional integral representation for the partitionfunction,Z = Tr e−β(H−µ·N)where µ = (µb, µf ) is the vector of chemical potentials, are also interested inthe operatorK := H − µ ·N = H0 + V − µ ·N. (5.6)102We can use the same argument as in the proof of Proposition 5.2.3 to showthat K is essentially self-adjoint on H(finite), provided that λb − µb > 0.Indeed, we assume that bothλb > µb and λf > µf . (5.7)We use these assumptions along with the bounds on H0 and V to find boundson the operator K.5.2.1 Bounds on KFrom Equations (5.1), (5.4), and (5.6),(λ− µ) ·N− Cv√Nb + 1 ≤ K ≤ (Λ− µ) ·N + Cv√Nb + 1. (5.8)For any γ > 0,Cv√Nb + 1 ≤ γ(Nb + 1) +C2vγ.Hence,(λ− µ) ·N− Cv√Nb + 1 ≥ (λb − µb − γ)Nb − γ −C2vγIf we take γ = λb−µb2 , then(λ− µ) ·N− Cv√Nb + 1 ≥λb − µb2Nb − κLwhereκL :=λb − µb2+C2vγ. (5.9)Thus, we have a lower bound for K that is purely linear in boson number,K ≥λb − µb2Nb − κL, (5.10)as well as the constant lower bound,K ≥ −κL. (5.11)1035.3 A Functional Integral Representationfor the Partition FunctionDefinition 14. The partition function isZ = Tr e−βKwhere β > 0 represents the inverse temperature.The goal in this section is to derive a functional representation for the par-tition function in the same style as (4.26):Tr e−βK = limq→∞∫ ∏τ∈Tq[dµR(q)(φ∗τ ,φτ )e−∫dy φ∗τ (y)·φτ (y)] ∏τ∈Tq〈φτ−|e−K |φτ 〉,where  := βq , Tq := {, 2, . . . , q} and∫dy φ∗τ (y) ·φτ (y) :=∫Xbdy φ∗b(y)φb(y) +∫Xfdy φ∗f (y)φf (y).The field φ0 is defined by the boundary condition,φ0 = φ˜β = (φb,−φf ).In order for the right-hand side to converge, the cutoff radius R(q) must growsufficiently quickly as q →∞. (We will see this in more detail later on.)In the case of particle-preserving interactions, we repeatedly used an ex-pansion over fixed-particle number subspaces in order to derive this type offunctional integral representation. In this case, the operator K does notpreserve bosons; as a result, the density operator e−βK may connect stateswith different boson numbers. That is, P (n)e−βKP (m) need not be zero whenn 6= m. However, if we letK0 := H0 − µ ·N,then K0 does preserve the number of bosons. The interaction V changesthe boson number by exactly one and we have bounds on P (n)V P (n±1). Wewill see that we can exploit these properties by expanding in powers of theinteraction V and then carrying out our analysis. This is called a Duhamel104expansion; we prove one version of this below.Theorem 5.3.1. (Duhamel’s Formula) For any ϕ, ψ ∈ H(finite), t > 0 andN ∈ N0,〈ϕ, e−tKψ〉 =N∑n=0An(t, ϕ, ψ) + N+1(t, ϕ, ψ)where for all n ∈ N,An(t, ϕ, ψ) := (−1)n∫Ωn(t)ds1 . . . dsn〈ϕ, e−(t−∑nk=1sk)K0[n∏k=1V e−skK0]ψ〉.(5.12)The domain of integration here isΩn(t) := {(s1, . . . , sn) ∈ [0, t]n :n∑k=1sk ≤ t}. (5.13)In the case n = 0, A0(t, ϕ, ψ) = 〈ϕ, e−tK0ψ〉. The “error term” is defined byn(t, ϕ, ψ) = (−1)n∫Ωn(t)ds1 . . . dsn〈ϕ, e−(t−∑nk=1sk)K[n∏k=1V e−skK0]ψ〉.(5.14)Proof. Fix ϕ and ψ in H(finite) and letJt(s) :=〈ϕ, e−(t−s)(K0+V )e−sK0ψ〉,for all s, t with 0 ≤ s ≤ t. Then Jt(0) = 〈ϕ, e−tKψ〉, Jt(t) = 〈ϕ, e−tK0ψ〉, andJ ′t(s) =〈ϕ,(e−(t−s)(K0+V )(K0 + V )e−sK0 − e−(t−s)(K0+V )e−sK0K0)ψ〉=〈ϕ, e−(t−s)(K0+V )V e−sK0ψ〉.Hence,〈ϕ, e−t(K0+V )ψ〉= Jt(0) = Jt(t)−∫ t0ds J ′t(s)=〈ϕ, e−tK0ψ〉−∫ t0ds〈ϕ, e−(t−s)(K0+V )V e−sK0ψ〉. (5.15)This is the N = 0 case of the claim. We proceed by induction. It suffices toshow that n(t, ϕ, ψ) = An(t, ϕ, ψ) + n+1(t, ϕ, ψ) for all n ≥ 1. We use the105same idea from the base case, applying it to the exponential function of Kthat appears in the definition of n(t, ϕ, ψ). That is,n(t, ϕ, ψ) = (−1)n∫Ωn(t)ds1 . . . dsn〈ϕ, e−(t−∑nk=1sk)K[n∏k=1V e−skK0]ψ〉= (−1)n∫Ωn(t)ds1 . . . dsn〈ϕ, e−(t−n+1∑k=1sk)Ke−sn+1K0[n∏k=1V e−skK0]ψ〉∣∣∣∣∣∣∣sn+1=0Let’s consider the right-hand side as a function f of sn+1 evaluated at sn+1 =0; that is, letf(s) := (−1)n∫Ωn(t)ds1 . . . dsn〈ϕ, e−(t−s−∑nk=1sk)Ke−sK0[n∏k=1V e−skK0]ψ〉.Thenn(t, ϕ, ψ) = f(0) = f(t−n∑k=1sk)−∫ t−n∑k=1sk0dsn+1 f′(sn+1).Here,f(t−n∑k=1sk)= (−1)n∫Ωn(t)ds1 . . . dsn〈ϕ, e−(t−n∑k=1sk)K0[n∏k=1V e−skK0]ψ〉= An(t, ϕ, ψ).The second term is−∫ t−n∑k=1sk0dsn+1 f′(sn+1)= −∫ t−n∑k=1sk0dsn+1∂∂sn+1(−1)n∫Ωn(t)ds1 . . . dsn〈ϕ, e−(t−∑n+1k=1sk)Ke−sn+1K0·[n∏k=1V e−skK0]ψ〉= (−1)n+1∫Ωn+1(t)ds1 . . . dsn+1〈ϕ, e−(t−∑n+1k=1sk)K(K −K0)e−sn+1K0106·[n∏k=1V e−skK0]ψ〉= (−1)n+1∫Ωn+1(t)ds1 . . . dsn+1〈ϕ, e−(t−∑n+1k=1sk)K[n+1∏k=1V e−skK0]ψ〉= n+1(t, ϕ, ψ).Hence,n(t, ϕ, ψ) = An(t, ϕ, ψ) + n+1(t, ϕ, ψ).This completes the proof.Our next step is to take the limit as N → ∞ in the Duhamel expansion ofTheorem 5.3.1 and prove that the resulting sum is absolutely convergent. Tothis end, we’ll need bounds on each term in the expansion, Ak(t, φ, ψ), aswell as the error term, k(t, φ, ψ), for large k.Lemma 5.3.2. Let t > 0 and n,m ∈ N0 be arbitrary.(a) For any M ∈ N, ψ ∈ H and φ ∈ Hn with ‖ψ‖ = ‖φ‖ = 1,|M(t, ψ, φ)| ≤(4CvCb max{t,√t})M√M !etκL , (5.16)where Cb := min{√λb − µb, 1}.(b) For any ψ ∈ Hm, φ ∈ Hn with ‖ψ‖ = ‖φ‖ = 1,|Ak(t, ψ, φ)| ≤ e−18 t(λb−µb)mC2(t)k√k!, (5.17)whereC2(t) := 4CvCbmax{t,√t}e14 t(λb−µb)=8|Xb||Xf |‖v‖∞min{√λb − µb, 1}max{t,√t}e14 t(λb−µb).(5.18)(c) For any ψ ∈ Hm, φ ∈ Hn with ‖ψ‖ = ‖φ‖ = 1,107|Ak(t, ψ, φ)| ≤C2(t)k(k!|m− n|!)1/4e−18 t(λb−µb)m. (5.19)Proof. (a) For any M ∈ N, ψ ∈ H and φ ∈ Hn with ‖ψ‖ = ‖φ‖ = 1,|M(t, ψ, φ)| =∣∣∣∣∣∫ΩM (t)ds1 . . . dsM〈ψ, e−(t−∑Mk=1sk)K[M∏k=1V e−skK0]φ〉∣∣∣∣∣≤∫ΩM (t)ds1 . . . dsM e(t−∑Mk=1sk)κL∥∥∥∥∥[M∏k=1V e−skK0]P (n)∥∥∥∥∥where −κL is the lower bound for K from Equation (5.11). Note thatK0 preserves boson number and satisfies the bounds(λ− µ) ·N ≤ K0 ≤ (Λ− µ) ·N. (5.20)Also, V changes the boson number by one. Hence,|M(t, ψ, φ)| ≤∫ΩM (t)ds1 . . . dsM e(t−∑Mk=1sk)κL·∑{nk}Mk=0nM=n(∀k)|nk−nk−1|=1M∏k=1∥∥∥P (nk−1)V P (nk)∥∥∥ e−sk(λb−µb)nk .From now on, we’ll understand that the sum is only over sequences{nk}Mk=0 with nM = n and |nk−nk−1| = 1 for all k ≥ 1, without writing allconditions explicitly. Using the bounds on∥∥∥P (n)V P (m)∥∥∥ from Equation(5.2) and bounding∑Mk=1 sk from below by zero,|M(t, ψ, φ)| ≤(Cv2)M ∫ΩM (t)ds1 . . . dsM etκL·∑{nk}M∏k=1[√nk + 1 e−sk(λb−µb)nk].We’ll use the exponential inside the product over k to cancel the factor108of√nk + 1. To this end, note that for any α > 0 and n ≥ 1,α(n+ 1) ≤ 2αn ≤ e2αn.=⇒√α(n+ 1) ≤ eαn=⇒ e−αn ≤1√α(n+ 1).Hence, if nk ≥ 1,√nk + 1 e−sk(λb−µb)nk ≤1√sk(λb − µb). (5.21)Otherwise nk = 0 and√nk + 1 e−sk(λb−µb)nk = 1. To account for bothcases, we simply let the product run only over k such that nk > 0. Thatis,|M(t, ψ, φ)| ≤( Cv2Cb)MetκL∫ΩM (t)ds1 · · · dsM∑{nk}M∏k=1nk>01√sk.where Cb := min{√λb − µb, 1}.We will show in the two lemmas after this proof that∫ΩM (t)ds1 . . . dsMM∏k=1nk>01√sk≤(4 max{t,√t})M√M !,a bound that it is independent of the sequence {nk}. Since all sequences{nk}Mk=1 must have nM = n, they can each be identified by their sequenceof differences ∆k = nk − nk−1 = ±1. Therefore, the number of allowedsequences {nk}Mk=0 is equal to 2M . Hence,|M(t, ψ, φ)| ≤(4CvCb max{t,√t})M√M !etκL .(b) Let ψ ∈ Hm, φ ∈ Hn with ‖ψ‖ = ‖φ‖ = 1.109Note that Ak(t, ψ, φ) and k(t, ψ, φ) differ only in the appearance of K0or K in the leftmost exponent. Since K0 is also bounded below (by zero),we can apply exactly the same analysis as above to obtain the bound|Ak(t, ψ, φ)| ≤(4CvCb max{t,√t})k√k!. (5.22)The right-hand side is the same as in (5.16), except with κL replaced by 0.However, we’ll also require a bound that diminishes as m,n→∞. Manyof the steps in the following calculation are the same as in the proof ofPart a, except that we use the assumption that ψ ∈ Hm (to project ontoHm in the second line below) and we’ll be more careful keep bounds thatdiminish as nk → 0 at each step.|Ak(t, ψ, φ)| =∣∣∣∣∣∣∫Ωk(t)ds1 . . . dsk〈ψ, e−(t−∑kj=1sj)K0k∏j=1V e−sjK0φ〉∣∣∣∣∣∣≤∫Ωk(t)ds1 . . . dsk e−(t−∑kj=1sj)(λb−µb)m∥∥∥∥∥∥P (m)k∏j=1V e−sjK0P (n)∥∥∥∥∥∥≤∫Ωk(t)ds1 . . . dsk e−(t−∑kj=1sj)(λb−µb)m·∑{nj}kj=0n0=m,nk=n(∀j)|nj−nj−1|=1k∏j=1(∥∥∥P (nj−1)V P (nj)∥∥∥∥∥∥P (nj)e−sjK0P (nj)∥∥∥)≤∫Ωk(t)ds1 . . . dsk e−(t−∑kj=1sj)(λb−µb)m∑{nj}k∏j=1(Cv2√nj + 1e−sj(λb−µb)nj)(5.23)In the last line, it is understood that the sequences {nj}kj=0 in the domainof the sum still must satisfy the conditions n0 = m, nk = n, and |nj −nj−1| = 1 for all j ≥ 1. Note that, in order for there to be any suchsequences, m+ n− k must be even, in which case these sequences mustsatisfynj ≥m+ n− k2,110for all j ∈ {1, . . . , k}. Consider the casek <m+ n2. (5.24)In this case, nj ≥ m+n4 . Thenk∏j=1e−sj(λb−µb)nj ≤k∏j=1(e−12 sj(λb−µb)nje−12 sj(λb−µb)m+n4)≤ e−18∑kj=1sj(λb−µb)(m+n)k∏j=1e−12 sj(λb−µb)njHence,|Ak(t, ψ, φ)| ≤∫Ωk(t)ds1 . . . dsk e−(t− 78∑kj=1sj)(λb−µb)m·∑{nj}k∏j=1(Cv2√nj + 1e−12 sj(λb−µb)nj)≤(Cv2)ke−18 t(λb−µb)m∫Ωk(t)ds1 . . . dsk∑{nj}k∏j=1(√nj + 1e−12 sj(λb−µb)nj)As in (5.21), when nk > 0,√nk + 1 e−12 sk(λb−µb)nk ≤1√12sk(λb − µb).Hence,|Ak(t, ψ, φ)| ≤(Cv2)ke−18 t(λb−µb)m∫Ωk(t)ds1 . . . dsk∑{nj}k∏j=1nj>0√2sk(λb − µb)≤(Cv√2Cb)ke−18 t(λb−µb)m∫Ωk(t)ds1 . . . dsk∑{nj}k∏j=1nj>01√sk111≤(Cv√2Cb)ke−18 t(λb−µb)m(4 max{t,√t})k√k!≤(4CvCb max{t,√t})k√k!e−18 t(λb−µb)m.Thus, in the case k < m+n2 ,|Ak(t, ψ, φ)| ≤(4CvCb max{t,√t})k√k!e−18 t(λb−µb)m. (5.25)Let’s now consider the case k ≥ m+n2 . Here, we can use the bound from(5.22),|Ak(t, ψ, φ)| ≤(4CvCb max{t,√t})k√k!.In the calculation below, let C1(t) := 4CvCb max{t,√t}. Since k ≥ m2 ,|Ak(t, ψ, φ)| ≤C1(t)k√k!≤C1(t)k√k!e−18 t(λb−µb)me−18 t(λb−µb)(2k)≤(C1(t)e14 t(λb−µb))k√k!e−18 t(λb−µb)m≤C2(t)k√k!e−18 t(λb−µb)m, (5.26)whereC2(t) := C1(t)e14 t(λb−µb) ≥ C1(t).From (5.25) for k ≤ m+n2 and (5.26) for k >m+n2 , we can conclude that,for all k ∈ N,|Ak(t, ψ, φ)| ≤C2(t)k√k!e−18 t(λb−µb)m.112(c) In part (b), note that |m− n| ≤ k in order for Ak(t, ψ, φ) to be nonzero.Hence,|Ak(t, ψ, φ)| ≤C2(t)k(k!|m− n|!)1/4e−18 t(λb−µb)m.Lemma 5.3.3. For all n ∈ N,In(t) :=∫Ωn(t)n∏k=11√skds1 . . . dsn ≤(4√t)n√n!,where, as above, Ωn(t) := {(s1, . . . , sn) ∈ [0, t]n :∑nk=1 sk ≤ t}.Proof. Make the substitution uj := sj/t for each j. Then,In(t) =∫Ωn(1)1∏nk=1√tuktn du1 . . . dun = tn/2In(1).In the case n = 1,I1(1) =∫ 101√sds = 2√s∣∣∣10= 2.For any n ∈ N,In+1(1) =∫Ωn+1(1)1∏n+1k=1√skds1 . . . dsndsn+1=∫ 10dsn+11√sn+1∫Ωn(1−sn+1)ds1 . . . dsn1∏nk=1√sk=∫ 10ds1√sIn(1− s)=∫ 10ds1√s(1− s)n/2In(1).113Now, for any  ∈ (0, 1),∫ 10ds(1− s)n/2√s=∫ 0ds(1− s)n/2√s+∫ 1ds(1− s)n/2√s≤∫ 0ds1√s+∫ 10ds(1− s)n/2√= 2√s∣∣∣0+1√[−(1− s)n/2+1n/2 + 1]10= 2√+1√·1n/2 + 1.If we take  = 1n+1 , then we find∫ 10ds(1− s)n/2√s≤2√n+ 1+√n+ 1n/2 + 1≤4√n+ 1.Hence,In+1(1) ≤4√n+ 1In(1).By induction, it follows thatIn(1) ≤4n−1√n!I1(1) <4n√n!.Hence,In(t) ≤(4√t)n√n!,as claimed.Lemma 5.3.4. For any n ∈ N and k ∈ N0 with 0 ≤ k ≤ n,Jn,k(t) :=∫Ωn(t)ds1 . . . dsnn∏j=k+11√sj≤4ntn+k2√(n+ k)!.Furthermore,Jn,k(t) ≤(4 max{t,√t})n√n!.114This second bound does not depend on k.Proof. The proof is by induction on k, with arbitrary n ∈ N satisfying n ≥ k.Lemma 5.3.3 is the case k = 0. Assuming that the claim holds for some kand all n ≥ k, then for any n ≥ k + 1 with n > 1,Jn,k+1(t) =∫Ωn(t)ds1 . . . dsnn∏j=k+21√sj=∫ t0ds1∫Ωn−1(t−s1)ds2 . . . dsnn∏j=k+21√sj=∫ t0ds Jn−1,k(t− s)≤∫ t0ds4n−1(t− s)n−1+k2√(n− 1 + k)!=4n−1√(n+ k − 1)!−(t− s)n+k+12n+k+12ts=0=2 · 4n−1√(n+ k − 1)!tn+k+12n+ k + 1≤4ntn+k+12√(n+ k + 1)!,as claimed. In the case n = 1, k must be equal to 0. ThenJn,k+1(t) = J1,1(t) =∫ t0ds 1 = t,which also satisfies the bound above.Lemma 5.3.5. For all m,n ∈ N0,‖P (m)e−tKP (n)‖ ≤√2eC2(t)2e−18 t(λb−µb)m. (5.27)115Furthermore, for any r > 1,‖P (m)e−tKP (n)‖ ≤e−18 t(λb−µb)m|m− n|!1/4(11− 1r4/3)3/4e14 (rC2(t))4. (5.28)Proof. For any m,n ∈ N0,‖P (m)e−tKP (n)‖ = supψ∈Hm, φ∈Hn‖ψ‖=‖φ‖=1∣∣∣〈ψ, e−tKφ〉∣∣∣ .From Lemma 5.3.2a, for any ψ ∈ Hm, φ ∈ Hn,limM→∞|M(t, ψ, φ)| = 0.Hence,∣∣∣〈ψ, e−tKφ〉∣∣∣ =∣∣∣∣∣∞∑k=0Ak(t, ψ, φ)∣∣∣∣∣≤∞∑k=0|Ak(t, ψ, φ)|≤∞∑k=0C2(t)k√k!e−18 t(λb−µb)m,where the last line is from Lemma 5.3.2b.Note that∞∑k=0C2(t)k√k!=∞∑k=0(1√2)k (√2C2(t))k√k!≤(∞∑k=0(12)k)1/2 ( ∞∑k=0(2C2(t)2)kk!)1/2≤√2eC2(t)2.Hence,‖P (m)e−tKP (n)‖ ≤√2eC2(t)2e−18 t(λb−µb)m,116as claimed in (5.27).To derive the tighter bound in (5.28), we use Lemma 5.3.2c.‖P (m)e−tKP (n)‖ = supψ∈Hm,φ∈Hn‖ψ‖=‖φ‖=1∣∣∣〈ψ, e−tKφ〉∣∣∣≤ supψ∈Hm,φ∈Hn‖ψ‖=‖φ‖=1∞∑k=0|Ak(t, ψ, φ)|≤∞∑k=0C2(t)k(k!|m− n|!)1/4e−18 t(λb−µb)m≤e−18 t(λb−µb)m|m− n|!1/4∞∑k=01rk(rC2(t))k(k!)1/4≤e−18 t(λb−µb)m|m− n|!1/4(∞∑k=01r43k)3/4 ( ∞∑k=0(rC2(t))4kk!)1/4≤e−18 t(λb−µb)m|m− n|!1/4(11− 1r4/3)3/4e14 (rC2(t))4.where r > 1 is an arbitrary constant.Theorem 5.3.6. For any t > 0, the partition function can be expanded asTr(e−tK) =∞∑n=0∑Y=(Yb,Yf )Yb∈Xnb /SnYf⊂Xf∞∑k=0Ak(t, δY , δY ), (5.29)where δY := δYb ⊗ δYf for each Y = (Yb, Yf ). Furthermore, the series above isabsolutely convergent.Proof. First, we prove that the sum on the right-hand side of Equation(5.29) is absolutely convergent. Noting that there are 2|Xf | subsets of Xf117and(|Xb|+n−1n)elements of Xnb /Sn and applying Lemma 5.3.2b,∞∑n=0∑Y=(Yb,Yf )Yb∈Xnb /SnYf⊂Xf∞∑k=0|Ak(t, δY , δY )|≤∞∑n=02|Xf |(|Xb|+ n− 1n)∞∑k=0e−18 t(λb−µb)nC2(t)k√k!≤ 2|Xf |(∞∑n=0(|Xb|+ n− 1)|Xb|−1(|Xb| − 1)!e−18 t(λb−µb)n)∞∑k=0C2(t)k√k!(5.30)<∞.Indeed, the sum converges absolutely.By Theorem 5.3.1, for any n,M ∈ N and Y = (Yb, Yf ) with Yb ∈ Xnb /Sn andYf ⊂ Xf ,〈δY , e−tKδY 〉 =M∑k=0Ak(t, δY , δY ) + M+1(t, δY , δY ).In the limit M →∞, M+1(t, δY , δY )→ 0, by Lemma 5.3.2a. Hence,〈δY , e−tKδY 〉 =∞∑k=0Ak(t, δY , δY ).Therefore,Tr(e−tK) =∞∑n=0∑Y=(Yb,Yf )Yb∈Xnb /SnYf⊂Xf〈δY , e−tKδY 〉=∞∑n=0∑Y=(Yb,Yf )Yb∈Xnb /SnYf⊂Xf∞∑k=0Ak(t, δY , δY ),as claimed.118Corollary 5.3.7. For any z ∈ C, let K(z) := K0 + zV . As a function of z,the partition function,Z(z) := Tr e−tK(z),is entire-analytic.Proof. Replacing all instances of V with zV in Theorem 5.3.6, we see thatthe right side of Equation (5.29) is a power series in z for Z(z),Z(z) =∞∑n=0∑Y=(Yb,Yf )Yb∈Xnb /SnYf⊂Xf∞∑k=0Ak(t, δY , δY )zk.Multiplying each term by zk in (5.30) does not influence the absolute con-vergence of the series, for any value of z ∈ C. We conclude that the seriesis absolutely convergent, uniformly convergent on compact subsets of C, andtherefore analytic on C.In the proof of the following theorem, we will use partial traces, which aredefined (in this context) as follows. Suppose that A is a trace-class operatoron H = B ⊗ F . If {φn}∞n=1 is any orthonormal basis for B, then the partialtrace of A over B isTrB(A) :=∞∑n=1〈φn, Aφn〉.Note that TrB(A) is an operator on F . Similarly, if {ψn}|Xf |n=1 is any basis forF , the partial trace of A over F isTrF(A) :=|Xf |∑n=1〈ψn, Aψn〉,an operator on B. Since {φi ⊗ ψj}i,j∈N is an orthonormal basis for B ⊗ F ,TrA =∞∑i,j=1〈φi ⊗ ψj, Aφi ⊗ ψj〉=∞∑i=1〈φi,∞∑j=1〈ψj, Aψj〉φi〉= TrB(TrF(A)).119Likewise,TrA = TrF(TrB(A)).Theorem 5.3.8. Let R be a map from N to (0,∞) satisfyinglimp→∞R(p)p√log p=∞.(For example, R(p) = p2 would work, as would any power of p that is strictlybigger than 1.) The partition function may be represented as the followingfunctional integral:Tr e−βK = limp→∞∫ ∏τ∈Tp[dµR(p)(φ∗τ ,φτ )e−∫dxφ∗τ (x)·φτ (x)] ∏τ∈Tp〈φτ−|e−K |φτ 〉,(5.31)where  := βp , Tp := {, 2, . . . , p}, and φ0 := φ˜β = (φβb ,−φβf ). In theexponent,∫dx φ∗τ (x) ·φτ (x) :=∫Xbdy φ∗b(y)φb(y) +∫Xfdy φ∗f (y)φf (y).Proof. We start by taking the partial trace of e−βK over the boson Fock spaceB. That is,TrB e−βK =∞∑n=0∑Y ∈Xnb /Sn〈δY , e−βKδY 〉.By the strong convergence of the bosonic resolution of the identity IR(p) tothe identity operator on B,TrB e−βK =∞∑n=0∑Y ∈Xnb /Snlimp→∞〈δY , e−βKIR(p)δY 〉.For all p > 0 and any n ∈ N0 and Y ∈ Xnb /Sn,∥∥∥〈δY , e−βKIR(p)δY 〉∥∥∥ ≤∥∥∥P (n)e−βKP (n)∥∥∥≤√2e−18β(λb−µb)n,120from the bound (5.27) in Lemma 5.3.5. Since there are(|Xb|+ n− 1n)≤(|Xb|+ n− 1)|Xb−1|(|Xb| − 1)!elements of Xnb /Sn,∞∑n=0∑Y ∈Xnb /Sn∥∥∥〈δY , e−βKIR(p)δY 〉∥∥∥ ≤ 1 +∞∑n=1(cn)|Xb−1|(|Xb| − 1)!√2eC2(β)2e−18β(λb−µb)n≤ 1 +√2eC2(β)2(|Xb| − 1)!∞∑n=1(cn)|Xb−1|e−18β(λb−µb)n<∞,where c is a constant that depends on |Xb|. By the dominated convergencetheorem, it follows thatTrB e−βK = limp→∞∞∑n=0∑Y ∈Xnb /Sn〈δY , e−βKIR(p)δY 〉= limp→∞TrB(e−βKIR(p)).By the approximate trace theorem for bosons (Proposition 3.6.1),TrB(e−βKIR(p)) =∫dµR(p)(φ∗, φ) e−∫Xbdy |φ(y)|2〈φ|e−βK |φ〉.We proceed by proving an intermediate version of Equation (5.31) that in-volves only the bosonic fields φτ ∈ L2(Xb),TrB(e−βK) = limp→∞∫ ∏τ∈Tp[dµR(p)(φ∗τ , φτ )e−‖φτ‖2] ∏τ∈Tp〈φτ−|e−K |φτ〉.To this end, we will show that the following difference converges to zero asp→∞:∥∥∥∥∥∥TrB(e−βKIR(p))−∫ ∏τ∈Tp[dµR(p)(φ∗τ , φτ )e−‖φτ‖2] ∏τ∈Tp〈φτ−|e−K |φτ〉∥∥∥∥∥∥≤∫dµR(p)(φ∗0, φ0)e−‖φ0‖2∥∥∥∥〈φ0|e−βK −(e−KIR(p))p−1e−K |φ0〉∥∥∥∥121=∫ ∏x∈Xbdφ∗0(x)dφ0(x)2piiχR(p)(|φ0(x)|)e−‖φ0‖2·∥∥∥∥〈φ0|e−βK −(e−KIR(p))p−1e−K |φ0〉∥∥∥∥ . (5.32)We will prove in Lemma 5.3.9 that, as long as limp→∞R(p)p√log p =∞,∥∥∥∥∥〈φ0|e−βK −(e−βpKIR(p))p−1e−βpK |φ0〉∥∥∥∥∥≤ 2|Xb|(16β(λb − µb))2e14(e−116β(λb−µb)+3)‖φ0‖2,for sufficiently large p.Therefore, the integrand of (5.32) satisfies the bound,χr(|φ0(x)|)e−‖φ0‖2∥∥∥∥〈φ0|e−βK −(e−KIR(p))p−1e−K |φ0〉∥∥∥∥≤ 4|Xb|(16β(λb − µb))2e− 14(1−e−116β(λb−µb))‖φ0‖2,which is independent of p and integrable with respect to the measure∏x∈Xbdφ∗0(x)dφ0(x)2pii.It follows by the dominated convergence theorem thatTrB(e−βK) = limp→∞∫ ∏τ∈Tp[dµR(p)(φ∗τ , φτ )e−‖φτ‖2] ∏τ∈Tp〈φτ−|e−K |φτ〉in operator norm.We now trace out the fermion space F . For clarity, we replace each “φ” inthe above expression with “φb”. Since F is finite-dimensional,Tr e−βK = limp→∞TrF∫ ∏τ∈Tp[dµR(p)(φ∗b,τ , φb,τ )e−‖φb,τ‖2] ∏τ∈Tp〈φb,τ−|e−K |φb,τ〉.122By the fermionic trace formula in Proposition 3.6.2,Tr e−βK = limp→∞∫dµ(φ∗0,f , φ0,f )e−∫Xfφ∗0,f (x)φf,0(x)〈−φf,0|U |φf,0〉.whereU :=∫ ∏τ∈Tp[dµR(p)(φ∗b,τ , φb,τ )e−‖φb,τ‖2] ∏τ∈Tp〈φb,τ−|e−K |φb,τ〉anddµ(φ∗f , φf ) :=∏x∈Xdφ∗f (x)dφf (x).We then apply the trace formula for fermions and insert the fermionic res-olution of the identity p times, each alongside an insertion of the bosonicapproximate resolution of the identity IR(p). Recall that the fermionic reso-lution of the identity is the Grassmann integral,1F =∫dµ(φ∗f , φf )e−∫Xfdy φ∗f (y)φf (y)|φf〉〈φf |,wheredµ(φ∗f , φf ) :=∏x∈Xfdφf (x)dφ∗f (x).Hence,Tr(e−βK) = limp→∞∫dµ(φ∗f,τ , φf,τ )e−∫Xfdy φ∗f,τ (y)φf,τ (y)· 〈−φf,τ |∫ ∏τ∈Tp[dµR(p)(φ∗b,τ , φb,τ )e−‖φb,τ‖2] ∏τ∈Tp〈φb,τ−|e−K |φb,τ〉|φf,τ 〉= limp→∞∫ ∏τ∈Tp[dµR(p)(φ∗b,τ , φb,τ )e−‖φb,τ‖2dµ(φ∗f,τ , φf,τ )e−∫Xfdy φ∗f,τ (y)φf,τ (y)]∏τ∈Tp〈φf,τ−|〈φb,τ−|e−K |φb,τ 〉|φf,τ 〉= limp→∞∫ ∏τ∈Tq[dµR(p)(φ∗τ ,φτ )e−∫dxφ∗τ (x)·φτ (x)] ∏τ∈Tq〈φτ−|e−K |φτ 〉,where φf,0 := −φf,τ . This completes the proof.The following lemma was used in the above proof.123Lemma 5.3.9. Let R be a map from N to (0,∞) satisfyinglimq→∞R(q)q√log q=∞ (5.33)For example, R(q) ∼= qs would satisfy this condition for any s > 1.Then,Tr e−βK = limq→∞Tr(e−βqKIR(q))q−1e−βqK . (5.34)Furthermore, for any φ ∈ L2(Xb), the operator on F ,〈φ|e−βK −(e−βqKIR(q))q−1e−βqK |φ〉,satisfies the bound,∥∥∥∥∥〈φ|e−βK −(e−βqKIR(q))q−1e−βqK |φ〉∥∥∥∥∥≤ 4|Xb|(16β(λb − µb))2e14(e−116β(λb−µb)+3)‖φ‖2L2(Xb) , (5.35)for sufficiently large q.Proof. For each i, q ∈ N, letLi :=e−βqK if i is oddIR(q) if i is evenand Mi :=e−βqK if i is odd1 if i is evenso that(e−βqKIR(q))q−1e−βqK =2q−1∏i=1Li and e−βK =2q−1∏i=1Mi.Noting that Li andMi differ only for even i, in which case, Li−Mi = IR(q)−1,(e−βqKIR(q))q−1e−βqK − e−βK =q−1∑`=12`−1∏j=1Li(IR(q) − 1)2q−1∏j=2`+1Mi .124Hence,∣∣∣∣∣Tr((e−βqKIR(q))q−1e−βqK − e−βK)∣∣∣∣∣=∣∣∣∣∣∣Trq−1∑`=12`−1∏j=1Li(IR(q) − 1)2q−1∏j=2`+1Mi∣∣∣∣∣∣≤∑n∈N0dimHn∥∥∥∥∥∥P (n)q−1∑`=12`−1∏j=1Li(IR(q) − 1)2q−1∏j=2`+1MiP (n)∥∥∥∥∥∥≤∑n∈N0dimHnq−1∑`=1∑{nj}qj=0n0=nq=nq∏j=1‖P (nj−1)e−βqKP (nj)‖∥∥∥(IR(q) − 1)P (n`)∥∥∥≤∑n∈N0dimHnq−1∑`=1∑{nj}qj=0n0=nq=nq−1∏j=0√2eC2()2e−18 (λb−µb)nj |Xb|e−R(q)2n∑`k=0R(q)2kk!(5.36)≤ (q − 1)e−R(q)2|Xb|(√2eC2()2)q(∞∑n=0dimHne−18 (λb−µb)n)·∞∑nj=0e−18 (λb−µb)njq−2∞∑n`=0e−18 (λb−µb)n`n∑`k=0R(q)2kk! .(5.37)where, in line (5.36) above, we’ve applied (5.27) from Lemma 5.3.5. As longas q > β,  < 1 andC2() =8|Xb||Xf |‖v‖∞min{√λb − µb, 1}max{,√}e14 (λb−µb)≤8|Xb||Xf |‖v‖∞min{√λb − µb, 1}e14 (λb−µb)√≤ c2√wherec2 :=8|Xb||Xf |‖v‖∞e14 (λb−µb)min{√λb − µb, 1}. (5.38)125The dimension of Hn satisfiesdimHn = dimF · dimBn= 2|Xf |(|Xb|+ n− 1n)≤ 2|Xf |(|Xb|+ n− 1)|Xb|−1(|Xb| − 1)!.If n ≥ 1,(|Xb|+ n− 1)|Xb|−1 ≤ (|Xb|n)|Xb|−1 ≤ |Xb||Xb|−1(|Xb| − 1)!α|Xb|−1eαn,where α is any positive constant. If we take α = 116(λb − µb),dimHn ≤2|Xf |(|Xb| − 1)!|Xb||Xb|−1(|Xb| − 1)!(116(λb − µb)βq)|Xb|−1e116 (λb−µb)βq n≤ 2|Xf ||Xb||Xb|−1(16q(λb − µb)β)|Xb|−1e116βq (λb−µb)n≤ C3 q|Xb|−1 e116βq (λb−µb)n, (5.39)whereC3 := 2|Xf |(16|Xb|(λb − µb)β)|Xb|−1.In the case n = 0, dimHn = 2|Xf |. Hence, the bound in (5.39) applies for alln ≥ 0 as long as q > (λb−µb)β16|Xb| .The first sum over n = n0 = nq in Equation (5.37) therefore satisfies∞∑n=0dimHne−18 (λb−µb)n ≤∞∑n=0C3q|Xb|−1e116 (λb−µb)ne−18 (λb−µb)n= C3q|Xb|−1∞∑n=0e−116 (λb−µb)n=C3q|Xb|−11− e−116 (λb−µb)126≤C3q|Xb|−1132(λb − µb)=32C3β(λb − µb)q|Xb|, (5.40)provided that q > β(λb−µb)16 . The sum over n` is∞∑n`=0e−18 (λb−µb)n`n∑`k=0R(q)2kk!=∞∑k=0R(q)2kk!∞∑n`=ke−18 (λb−µb)n`=∞∑k=0R(q)2kk!e−18 (λb−µb)k1− e−18 (λb−µb)=ee− 18 (λb−µb)R(q)21− e−18 (λb−µb)≤ee− 18 (λb−µb)R(q)2116(λb − µb)≤16β(λb − µb)qee− 18 (λb−µb)R(q)2 . (5.41)We’ve assumed in the fourth line that q is large enough so that 18βq (λb−µb) <1, i.e. q > 18β(λb − µb) The sum over nj, 0 < j < q, j 6= ` is∞∑nj=0e−18 (λb−µb)nj =11− e−18 (λb−µb)≤1116(λb − µb)=16qβ(λb − µb). (5.42)Applying each of the bounds from (5.40), (5.41), and (5.42) to Equation(5.37),∣∣∣∣∣Tr((e−βqKIR(q))q−1e−βqK − e−βK)∣∣∣∣∣≤ qe−R(q)2|Xb|(√2eC2()2)q(∞∑n=0dimHne−18 (λb−µb)n)127·∞∑nj=0e−18 (λb−µb)njq−2∞∑n`=0e−18 (λb−µb)n`n∑`k=0R(q)2kk!≤ qe−R(q)2|Xb|2q/2ec22β32C3β(λb − µb)q|Xb|(16qβ(λb − µb))q−2 16qee− 18 (λb−µb)R(q)2β(λb − µb)≤ 2C3|Xb|q|Xb|ec22β(16√2qβ(λb − µb))qe−(1−e− 18 (λb−µb))R(q)2≤ 2C3|Xb|ec22βe−116β(λb−µb)R(q)2q +q log(c3q)+|Xb| log q,where c3 := 16√2β(λb−µb). We’ve assumed that q is larger than both β and18β(λb − µb).This converges to zero if R(q) → ∞ sufficiently quickly, for example, ifR(q) ∼ q3/2 for large q. More generally, if R satisfies the conditionlimq→∞R(q)q√log q=∞as assumed in (5.33), we may conclude that∣∣∣∣∣Tr((e−βqKIR(q))q−1e−βqK − e−βK)∣∣∣∣∣→ 0as q →∞. SinceIR(q) = IR(q) ⊗ 1F ,this completes the proof of the first statement (5.34) in the lemma. We nowmove on to prove the second statement, (5.35).For any φ ∈ L2(Xb), 〈φ|e−βK −(e−βqKIR(q))q−1e−βqK |φ〉 is an operator onF . In order to find a bound on the norm of this operator, we will applythe same ideas from the proof above, except that we will use the bound(5.28) of Lemma 5.3.5, which is tighter than (5.27). The resulting boundon∥∥∥∥∥〈φ|e−βK −(e−βqKIR(q))q−1e−βqK |φ〉∥∥∥∥∥will allow for the application of the128dominated convergence theorem in the proof of Theorem 5.3.8.∥∥∥∥∥〈φ|e−βK −(e−βqKIR(q))q−1e−βqK |φ〉∥∥∥∥∥≤∑m,n∈N0‖φ‖n+m√n!m!∥∥∥∥∥P (n)(e−βK −(e−βqKIR(q))q−1e−βqK)P (m)∥∥∥∥∥≤∑m,n∈N0‖φ‖n+m√n!m!∥∥∥∥∥∥P (n)q−1∑`=12`−1∏j=1Li(IR(q) − 1)2q−1∏j=2`+1MiP (m)∥∥∥∥∥∥≤∑m,n∈N0‖φ‖n+m√n!m!q−1∑`=1∑{nj}qj=0n0=n,nq=mq∏j=1‖P (nj−1)e−βqKP (nj)‖∥∥∥(IR(q) − 1)P (n`)∥∥∥≤∑m,n∈N0‖φ‖n+m√n!m!q−1∑`=1∑{nj}qj=0n0=n,nq=mq−1∏j=0e−18 (λb−µb)nj|nj+1 − nj|!1/4(11− 1r4/3)3/4e14 (rC2())4|Xb|e−R(q)2n∑`k=0R(q)2kk!≤∑m,n∈N0‖φ‖n+m√n!m!e−18β(λb−µb)n∑{nj}qj=0n0=n,nq=mq−1∑`=1|Xb|e−R(q)2n∑`k=0R(q)2kk!·q−1∏j=0e−18 (λb−µb)(nj−n)|nj+1 − nj|!1/4(11− 1r4/3)3/4e14 (rC2())4≤ |Xb|e−R(q)2(11− 1r4/3) 34 qeq4 (rC2())4 ∑m,n∈N0‖φ‖n+m√n!m!e−18β(λb−µb)n·q−1∑`=1∑{nj}qj=0n0=n,nq=mn∑`k=0R(q)2kk!q−1∏j=0e−18 (λb−µb)(nj−n)|nj+1 − nj|!1/4≤ |Xb|e−R(q)2(11− 1r4/3) 34 qeq4 (rC2())4 ∑m,n∈N0‖φ‖n+m√n!m!e−18β(λb−µb)nq−1∑`=1∞∑n′`=0129e−18 (λb−µb)(n′`−n)n′∑`k=0R(q)2kk!∑{nj}qj=0n0=n,nq=me18 (λb−µb)(n`−n)q−1∏j=0(e−8 (λb−µb)(nj−n)|nj+1 − nj|!1/4)(5.43)where r > 1 is an arbitrary constant.Bounding the sum over n′`,∞∑n′`=0e−18 (λb−µb)(n′`−n)n′∑`k=0R(q)2kk!= e18 (λb−µb)n∞∑n′`=0e−18 (λb−µb)n′`n′∑`k=0R(q)2kk!= e18 (λb−µb)n∞∑k=0R(q)2kk!∞∑n′`=ke−18 (λb−µb)n′`= e18 (λb−µb)n∞∑k=0R(q)2kk!e−18 (λb−µb)k1− e−18 (λb−µb)= e18 (λb−µb)n11− e−18 (λb−µb)ee− 18 (λb−µb)R(q)2 .(5.44)Next, we bound the sum over the sequences {nj : j ∈ {0, . . . , q}, n0 = n, nq =m} that appears as the rightmost sum in (5.43). For each j ∈ {1, · · · , q}, letmj := nj − nj−1.∑{nj}qj=0n0=n,nq=me+18 (λb−µb)(n`−n)q−1∏j=0e−18 (λb−µb)(nj−n)|nj+1 − nj|!1/4≤∑{nj}qj=0n0=n,nq=mq−1∏j=0e18 (λb−µb)|nj−n||nj+1 − nj|!1/4≤∑{mj}qj=1∈Zq∑qj=1mj=m−nq−1∏j=0e18 (λb−µb)|∑ji=1mi||mj+1|!1/4130≤∑{mj}qj=1∈Zq∑jmj=m−nq∏j=1e18 (λb−µb)∑j−1i=1|mi||mj|!1/4=∑{mj}qj=1∈Zq∑jmj=m−nq∏j=1e18 (λb−µb)(q−j)|mj ||mj|!1/4≤ 2qq∏j=1∞∑mj=0e18 (λb−µb)(q−j)mjmj!1/4.Note that mq is determined by m,n and the other mj, but we summed overit anyway. By Hölder’s inequality,∞∑m=0e18 (λb−µb)(q−j)mm!1/4=∑m∈Z1r′mr′me18 (λb−µb)(q−j)mm!1/4≤(∞∑m=01r′43m)3/4∞∑m=0(r′4e12 (λb−µb)(q−j))mm!1/4≤(11− r′−43)3/4 (er′4e12 (λb−µb)(q−j))1/4,where r′ > 1 is an arbitrary constant. Hence,∑{nj}qj=0n0=n,nq=me+18 (λb−µb)(n`−n)q−1∏j=0e−18 (λb−µb)(nj−n)|nj+1 − nj|!1/4≤(2(1− r′−43 )34)q q∏j=1er′44 e12 (λb−µb)(q−j)≤(2(1− r′−43 )34)qer′44∑qj=1e12 (λb−µb)(q−j)=(2(1− r′−43 )34)qer′44∑q−1j=0e12 (λb−µb)j=(2(1− r′−43 )34)qexpr′44e12 q(λb−µb) − 1e12 (λb−µb) − 1 . (5.45)131Applying the bounds in (5.44) and (5.45) to (5.43),∥∥∥∥∥〈φ|e−βK −(e−βqKIR(q))q−1e−βqK |φ〉∥∥∥∥∥≤ |Xb|e−R(q)2(11− 1r4/3) 34 qeq4 (rC2())4 ∑m,n∈N0‖φ‖n+m√n!m!e−18β(λb−µb)nq·(2(1− r′−43 )34)qexpr′44e12 q(λb−µb) − 1e12 (λb−µb) − 1 e18 (λb−µb)n11− e−18 (λb−µb)· exp(e−18 (λb−µb)R(q)2)(5.46)≤ |Xb|(2(1− r−43 )32)qeq4 (rC2())4 ∑m,n∈N0‖φ‖n+m√n!m!e−18 (β−)(λb−µb)nq· expr44e12β(λb−µb) − 114(λb − µb)1116(λb − µb)e−116 (λb−µb)R(q)2≤ |Xb|16β(λb − µb)e− 116β(λb−µb)R(q)2q +14 (rc2)4 β2q +(r4 e12β(λb−µb)β(λb−µb)+c3)q+2 log q·∑m,n∈N0‖φ‖n+m√n!m!e−18 (β−)(λb−µb)n, (5.47)wherec3 := log(2(1− r−43 )32).In the second inequality, we’ve set the arbitrary constants r and r′ equal toeach other. We’ve also assumed that q is larger than both 18β(λb − µb) andβ, so that1− e18 (λb−µb) ≥116(λb − µb).andC2() ≤ c2 max{,√} ≤ c2√ = c2√βq,where c2 was defined in (5.38) to bec2 =8|Xb||Xf |‖v‖∞e14 (λb−µb)min{√λb − µb, 1}.132Let s > 1 be an arbitrary constant.∞∑m=0‖φ‖m√m!=∞∑m=01sm/2(√s‖φ‖)m√m!≤(∞∑m=01sm)1/2 ( ∞∑m=0sm‖φ‖2mm!)1/2≤( ss− 1)1/2e12 s‖φ‖2. (5.48)Similarly,∞∑n=0(‖φ‖e−18 (β−)(λb−µb))n√n!≤( ss− 1)1/2e12 se− 14 (β−)(λb−µb)‖φ‖2 . (5.49)Applying these inequalities (5.48) and (5.49) to the main bound in (5.47),∥∥∥∥∥〈φ|e−βK −(e−βqKIR(q))q−1e−βqK |φ〉∥∥∥∥∥≤ |Xb|16β(λb − µb)e− 116β(λb−µb)R(q)2q +14 (rc2)4eβ(λb−µb)q β2q +(r4 e12β(λb−µb)β(λb−µb)+c3)q+2 log q·( ss− 1)e12 s(e−14 (β−)(λb−µb)+1)‖φ‖2≤ |Xb|16β(λb − µb)e− 116β(λb−µb)R(q)2q +14 (rc2)4eβ(λb−µb)q β2q +(r4 e12β(λb−µb)β(λb−µb)+c3)q+2 log q·( ss− 1)e12 s(e−18β(λb−µb)+1)‖φ‖2,as long as q > 2.If we lets =12(1 +2e−18β(λb−µb) + 1),then12s(e−18β(λb−µb) + 1)=12(12+1e−18β(λb−µb) + 1)(e−18β(λb−µb) + 1)133=14(e−18β(λb−µb) + 3).Also,ss− 1=3 + e−18β(λb−µb)1− e−18β(λb−µb)≤4 · 16β(λb − µb).Since limq→∞R(q)q√log q =∞,e− 116β(λb−µb)R(q)2q +14 (rc2)4eβ(λb−µb)q β2q +(r4 e12β(λb−µb)β(λb−µb)+c3)q+2 log q→ 0as q →∞. If q is sufficiently large, this is less than 1, and∥∥∥∥∥〈φ|e−βK −(e−βqKIR(q))q−1e−βqK |φ〉∥∥∥∥∥≤ 4|Xb|(16β(λb − µb))2e14(e−116β(λb−µb)+3)‖φ‖2,as claimed.The following lemma was used in the proof above.Lemma 5.3.10. For any r > 0 and n ∈ N,‖(1− Ir)Pn‖ ≤ |Xb|e−r2n∑k=0r2kk!Proof. Recall that, from [10] (pg. 1246), for any r > 1,‖(1− Ir)Pn‖ ≤ |Xb|max|Y |≤nmaxx∈Xb1µY (x)!∫ ∞r2dt e−ttµY (x)= |Xb|max|Y |≤nmaxx∈Xbe−r2µY (x)∑k=0r2kk!≤ |Xb|e−r2n∑k=0r2kk!.134The second line can easily be proven by induction on µY (X), as we now show.The base case is the equation∫∞r2 e−tdt = e−r2. The induction step is∫ ∞r2e−ttnn!dt = − e−ttnn!∣∣∣∣∞r2+∫ ∞r2e−ttn−1(n− 1)!dt= e−r2 r2nn!+ e−r2n−1∑k=0r2kk!= e−r2n∑k=0r2kk!.This completes the proof of Theorem 5.3.8.5.4 The ActionWe define the action F (,α∗,φ) to be logarithm of the inner product whichappeared in the functional integral (5.31),F (,α∗,φ) := log〈α|e−K |φ〉,wherever this logarithm exists. We will see in this short section that F (, ·, ·)is an entire-analytic function of the fields α∗ = (α∗b , α∗f ) and φ = (φb, φf ).Proposition 5.4.1. The inner product,G(,α∗,φ) := 〈α|e−K |φ〉is an entire-analytic nonzero function of the fermion fields, α∗f (y) and φf (y),for all y ∈ Xf .Proof. For each n ∈ N, let P (n)f be the projection onto the n-fermion sub-space of H, B ⊗ Fn.135Since K preserves fermion number,G(,α∗,φ) = 〈α|e−K |φ〉=|Xf |∑n=0〈α|P (n)f e−KP (n)f |φ〉=|Xf |∑n=0〈αf |P(n)f[〈αb|P(n)f e−KP (n)f |φb〉]P (n)f |φf〉, (5.50)noting that each 〈αb|P(n)f e−KP (n)f |φb〉 is an operator on Fn.For each n,P (n)f |φf〉 =∑Y⊂Xf|Y |=n〈δY |φf〉δY=∑y1,...,yn∈Xfy1<···<ynφf (y1) · · ·φf (yn)δ{y1,...,yn}. (5.51)Similarly,〈αf |P(n)f =∑Y⊂Xf|Y |=n〈αf |δY 〉δ∗Y=∑y1,...,yn∈Xfy1<···<ynα∗f (yn) · · ·α∗f (y1)δ∗{y1,...,yn}. (5.52)where δ∗Y is the dual of δY . From Equations (5.50), (5.51), and (5.52),G(,α∗,φ) is the following polynomial in the fermion fields φf and α∗f ,G(,α∗,φ) =|Xf |∑n=0∑y1<···<ynx1<···<xn〈δ{y1,...,yn},[〈αb|P(n)f e−KP (n)f |φb〉]δ{x1,...,xn}〉· α∗f (yn) · · ·α∗f (y1)φf (x1) · · ·φf (xn).Hence, G(,α∗,φ) is an analytic function of the fermion fields, φf and α∗f .To see that G(,α∗,φ) is nonzero everywhere, it suffices to prove that the136zeroth-order term in the fermion fields, G0(,α∗,φ) := 〈αb|P(0)f e−KP (0)f |φb〉,is nonzero everywhere. This is done in Lemma 5.4.2 below.Lemma 5.4.2. In the expansion forG(,α∗,φ) = 〈α|e−K |φ〉 in the fermionfields α∗f and φf , the zeroth-order term, G0(,α∗,φ), is an entire-analyticfunction of the boson fields α∗b and φb and is non-zero everywhere. Moreover,G0(,α∗,φ) = exp(∫X2bdx dy α∗b(x)e−(hb−µb)(x, y)φb(y)). (5.53)Proof. First note thatG0(,α∗,φ) = 〈α|P(0)f e−KP (0)f |φ〉= 〈αb|e−(Hb−µbNb)|φb〉,whereHb :=∫X2bdx dy a†b(x)hb(x, y)ab(y).Also,Hb − µbNb =∫X2bdx dy a†b(x)(hb(x, y)− µb)ab(y).From Lemma 2.23 of [10],〈αb|e−(Hb−µbNb)|φb〉 = 〈αb|e−(hb−µb)φb〉= exp(〈αb, e−(hb−µb)φb〉L2(Xb))= exp(∫X2bdx dy α∗b(x)e−(hb−µb)(x, y)φb(y)),as claimed. Indeed, this is an entire-analytic function of the boson fields, α∗band φb, that is nonzero everywhere.Proposition 5.4.3. The action,F (,α∗,φ) = log〈α|e−K |φ〉,is an entire-analytic function of the fields α∗ and φ.Proof. For each n ∈ {0, . . . , |Xf |}, let Gn(,α∗,φ) be the component of137G(,α∗,φ) of order n in the fermion fields,Gn(,α∗,φ) := 〈α|P(n)f e−KP (n)f |φ〉.Note that each Gn(,α∗,φ) is an entire-analytic function of the fields α∗ andφ. Then,G(,α∗,φ) =|Xf |∑n=0Gn(,α∗,φ).From Lemma 5.4.2, G0(,α∗,φ) is nonzero everywhere and may be factoredout of the sum,G(,α∗,φ) = G0(,α∗,φ)1 +|Xf |∑n=1Gn(,α∗,φ)G0(,α∗,φ) .Taking the logarithm,F (,α∗,φ) = logG(,α∗,φ)= logG0(,α∗,φ) + log1 +|Xf |∑n=1Gn(,α∗,φ)G0(,α∗,φ) , (5.54)wherelogG0(,α∗,φ) =∫X2bdx dy α∗b(x)e−(hb−µb)(x, y)φb(y),by Equation (5.53). Since each Gn(,α∗,φ) is of order at least 1 in theGrassmann algebra, the second term in (5.54) is given by the Taylor polyno-mial,log1 +|Xf |∑n=1Gn(,α∗,φ)G0(,α∗,φ) =|Xf |∑m=11m|Xf |∑n=1Gn(,α∗,φ)G0(,α∗,φ)m=|Xf |∑m=11me−m∫X2bdx dy α∗b (x)e−(hb−µb)(x,y)φb(y)|Xf |∑n=1Gn(,α∗,φ)m.138Hence,F (,α∗,φ) =∫X2bdx dy α∗b(x)e−(hb−µb)(x, y)φb(y)+|Xf |∑m=11me−m∫X2bdx dy α∗b (x)e−(hb−µb)(x,y)φb(y)|Xf |∑n=1Gn(,α∗,φ)m.(5.55)Since each Gn(,α∗,φ) is an entire-analytic function of the fields α∗ and φ,it follows that F (,α∗,φ) is as well.Hence, in the functional integral for the partition function (5.31) of Theorem5.3.8, we can replace each inner product 〈φτ−|e−K |φτ 〉 with the exponentialof the action F (,φ∗τ−,φτ ) to obtain,Tr e−βK = limp→∞∫ ∏τ∈Tp[dµR(p)(φ∗τ ,φτ )e−∫dxφ∗τ (x)·φτ (x)] ∏τ∈TpeF (,φ∗τ−,φτ ).(5.56)139Chapter 6ConclusionWe have rigorously constructed functional integral representations for twotypes of quantum many-particle systems. The first system consists of amixed gas with a finite number of particle species, bosons and/or fermions,interacting on a finite lattice with a general Hamiltonian that preserves thetotal number of particles in each species. Our first main result is the coherent-state functional integral representation for the partition function,Tr e−βK = limq→∞∫ ∏τ∈Tq[dµR(q)(φ∗τ ,φτ )e−∫dy φ∗τ (y)·φτ (y)] ∏τ∈Tq〈φτ−|e−K |φτ 〉,(6.1)in Theorem 4.2.1. Here, R(q) is a large-boson-field cutoff which growssufficiently quickly as q → ∞. In Section 4.3, we expanded the “action”F (,α∗,φ) := log 〈α|e−K |φ〉 in powers of the fields α and φ and deriveda recursion relation for the coefficients. In the case of a 2-body interaction,such as the Coulomb interaction, we proved bounds on these coefficients andused them to find a domain of analyticity for the action. This domain islarge enough that eF (,α∗,φ) may be substituted for each inner product of theform 〈α|e−K |φ〉 in the functional integral (6.1) above, as long as the large-field-cutoffs R(q) are taken to be not too large.The second system consists of one species of bosons and one species offermions interacting on a finite lattice, which can be thought of as phononsand electrons respectively. The Hamiltonian takes a specific form, whereinthe interaction term does not preserve the number of bosons. In this case,we generally use a Duhamel formula to expand in powers of the interaction,140rather than over particle numbers. In Theorem 5.3.8, we find a rigorouscoherent-state functional integral representation for the partition function,which is also of the form (6.1) above. Finally, we use properties of the Grass-mann algebra in Section 5.4 to show that the “action” is an entire-analyticfunction of the fields.141Bibliography[1] M. Reed and B. Simon, Methods of Modern Mathematical Physics,Volume 2: Fourier Analysis, Self-Adjointness, Academic Press, NewYork (1972). → pages 2, 4, 101[2] M. Reed and B. Simon, Methods of Modern Mathematical Physics,Volume 1: Functional Analysis, Academic Press, New York (1972). →pages 2, 11, 13[3] S. Albeverio, R. Høegh-Krohn, and S. Mazzucchi, Mathematical Theoryof Feynman Path Integrals, Lecture Notes in Mathematics, 523,Springer (2008). → pages 3[4] J.L. Doob, Wiener’s Work in Probability Theory, Bulletin of theAmerican Mathematical Society, 72 (1, Part II), 69-72 (1966). → pages4[5] P. Moral, Feynman-Kac Formulae - Genealogical and InteractingParticle Systems with Applications, Springer, New York (2004). →pages 4[6] J. Ginibre, Reduced Density Matrices of Quantum Gases. I: Limit ofInfinite Volume, Journal of Mathematical Physics, 4, 238-251 (1965). →pages 5[7] P. Federbush and D. Brydges, The cluster expansion in statisticalmechanics, Communications in Mathematical Physics, 49, 233-246(1976). → pages 5[8] M. I. Freidlin, Functional Integration and Partial Differential Equations,Princeton University Press, Princeton, NJ (1985). → pages 5142[9] J.W. Negele and H. Orland, Quantum Many-Particle Systems,Addison-Wesley, Boulder, CO (1988). → pages 5, 8, 32[10] T. Balaban, J. Feldman, H. Knörrer, and E. Trubowitz, A functionalintegral representation for many Boson systems. I: The PartitionFunction, Annales Henri Poincaré, 9, 1229-1273 (2008). → pages 5, 7,8, 32, 47, 52, 53, 63, 65, 69, 88, 89, 134, 137[11] S. Ospelkaus, C. Ospelkaus, L. Humbert, K. Sengstock, and K. Bongs,Tuning of Heteronuclear Interactions in a Degenerate Fermi-BoseMixture, Physical Review Letters, 97, 120403 (2006). → pages 7[12] H.P. Büchler and G. Blatter, Supersolid versus Phase Separation inAtomic Bose-Fermi Mixtures, Physical Review Letters, 91, 13 (2003). →pages[13] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, A. Sen(De) andU. Sen, Ultracold atomic gases in optical lattices: mimicking condensedmatter physics and beyond, Advances in Physics, 56:2, 243-379 (2007).→ pages 7[14] T. Holstein, Studies of Polaron Motion, Part I. The Molecular-CrystalModel, Annals of Physics, 281, 706-724 (2000). → pages 7, 95[15] F. Marsiglio and J.P. Carbotte, Electron-Phonon Superconductivity,Superconductivity - Conventional and Unconventional Superconductors,Volume 1, Springer-Verlag Berlin Heidelberg (2008). → pages 7, 95[16] J. Hubbard, Calculation of Partition Functions, Physical ReviewLetters, 3, 77-78 (1959). → pages 8[17] S.J. Gustafson and I.M. Sigal, Mathematical Concepts of QuantumMechanics, Springer-Verlag Berlin Heidelberg (2003). → pages 11, 13[18] R.F. Streater and A.S. Wightman, PCT, Spin and Statistics, and AllThat, Princeton University Press, Princeton, NJ, 146 - 161 (2000). →pages 15[19] J. Feldman, H. Knörrer, and E. Trubowitz, Fermionic FunctionalIntegrals and the Renormalization Group, CRM Monograph Series 16,American Mathematical Society (2002). → pages 38, 39, 40143[20] A. Fetter and J. Walecka, Quantum Theory of Many-Particle Systems,McGraw-Hill, New York (1971). → pages 95144

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