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The quartic interaction in the large-N limit of quantum field theory on a noncummutative space DeBoer, Philip Albert 2001

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THE  QUARTIC INTERACTION  IN T H E L A R G E - N L I M I T O F  Q U A N T U M FIELD T H E O R Y ON A N O N C O M M U T A T I V E By Philip Albert DeBoer B . Sc. University of Prince Edward Island, 1999  A THESIS SUBMITTED  IN PARTIAL FULFILLMENT OF  THE REQUIREMENTS  FOR T H EDEGREE OF  M A S T E R OF  SCIENCE  in THE FACULTY OF GRADUATE DEPARTMENT  O F PHYSICS  AND  STUDIES ASTRONOMY  We accept this thesis as conforming to the required standard  THE UNIVERSITY  O F BRITISH  COLUMBIA  August 2001 ©  Philip Albert DeBoer, 2001  SPACE  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study.  I further agree that permission for extensive copying of this  thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.  It is understood that copying or publication of this thesis for  financial gain shall not be allowed without my written permission.  Department of Physics and Astronomy T h e University of British Columbia 6224 Agricultural Road Vancouver, B . C . , Canada V 6 T 1Z1  Date:  Abstract W i t h a view to understanding generic properties of quantum field theories defined on spaces with noncommuting spatial coordinates (termed noncommutative quantum field theories), two simple models are considered. T h e first model is a theory of bosonic vector fields having an 0(iV)-symmetric quartic interaction. T h e second model is the fermionic counterpart of the bosonic theory, and is known as the Gross-Neveu model. In both cases the study is conducted in the simplifying large-TV limit. Unlike in the commutative case, the noncommutative theory gives rise to two inequivalent quartic interactions of the form (</> ) and (<j) <f)i) '. T h e latter interaction is difficult 2 2  %  2  to work with, but significant progress is made for the theories containing only the former interaction.  ii  Table of Contents  Abstract  ii  Table of Contents  iii  Acknowledgments  v  1  Introduction  1  1.1  Motivation  1  1.2  Q u a n t u m Field Theory  4  1.2.1  Saddle-Point Approximation  5  1.2.2  Effective Action  7  1.2.3  Large-iV  8  1.2.4  Renormalization  1.3  1.4  2  10  Moyal Product  10  1.3.1  Construction  11  1.3.2  Properties  14  Noncommutative Q u a n t u m Field Theories  14  Noncommutative O(N) 0 Theory  16  2.1  Introduction  16  2.2  Local Auxiliary Field  18  2.3  Two-point Propagator  19  2.4  Self-Energy  20  4  iii  2.5  2.6  3  22  2.5.1  26  Four Dimensions  Beyond the Classical Case  Noncommutative  27  Gross-Neveu Theory  32  3.1  Introduction  32  3.2  Local Auxiliary Field  33  3.3  Perturbation A b o u t the Classical Solution  33  3.4  T w o Dimensions  35  3.4.1  T h e Commutative Theory  35  3.4.2  T h e Noncommutative Theory  36  3.4.3  A Double-Scaling Limit  38  3.5  4  Symmetric Vertex  Three Dimensions  38  3.5.1  T h e Commutative Theory  39  3.5.2  Noncommutative Theory  40  Conclusions  44  Bibliography  46  iv  Acknowledgments Thanks are due to my supervisor D r . G . Semenoff for his guidance and insightful comments during the course of this work. I would also like to thank E m i l for his willingness to answer all my little questions. Thanks also to M a r k and Ben for their encouragement. Finally, I would like to thank Drs.  J . M c K e n n a and especially K . Schleich for their  support during the final stages of this work.  Great is our L o r d and mighty in power; his understanding has no limit. Psalm 147:5 NIV  Chapter 1  Introduction  1.1  Motivation  Recently there has been much interest in noncommutative quantum field theories. T h e low-energy limit of string theories with background antisymmetric tensor fields [1, 2, 3, 4, 5, 6] gives rise to certain noncommutative quantum field theories. T h e y retain some of the interesting features of string theory such as nonlocality, which can be studied in the more familiar context of quantum field theory. Since the string theories are consistent quantum mechanical theories, the noncommutative field theories which are their consistent zero slope limits should also be internally consistent, that is, unitary and renormalizable. In fact, for some theories, unitarity has been demonstrated explicitly at one-loop order [7]. Because the theory generated by the string theory limit is difficult to study, simpler quantum field theories are being studied in the noncommutative regime. T h e field theories which arise from string theory are the Yang-Mills theories; these quantum fields have a nonabelian gauge symmetry. These theories are technically difficult to study; one of the problems is a lack of local gauge-invariant observables in the noncommutative case. A s a result, in order to better understand generic features of these theories, simpler examples have been chosen for study. In particular, the theories studied here are theories of N fields with an O(N)  symmetry and having quartic interactions.  B o t h bosonic and fermionic models will be examined. These models have the advantage of being solvable in the large-N limit, but since these theories do not arise as string  Chapter 1.  Introduction  theory limits there is no reason a priori  2  to assume internal consistency.  T h e concept of field theory on noncommutative geometry is not entirely new, however. Because quantum field theories are plagued by divergences arising from short-distance behaviour, noncommutativity has been suggested as a way to tame these divergences. T h e argument is that the uncertainty relation can be used to establish a short-distance cutoff so that distances shorter than the cutoff can be interpreted as long distances.  It  was shown by F i l k [15] that this need not be the case. For the theories considered here it will be seen that although the noncommutativity partially reduces the short-distance ultraviolet divergences, new divergences appear in the long-distance infrared regime. T h e remainder of this chapter introduces the important techniques required for studying noncommutative quantum field theories. In Chapter 2 the noncommutative O ( N ) c/> theory is studied in the large-N limit. 4  In the critical dimension the coupling is dimensionless and the theory is marginally renormalizable by power counting. Here this occurs in four dimensions, and this is where the theory is studied in this paper. Although two possible interactions exist in the noncommutative case, only the symmetric vertex will be examined in detail. T h i s allows the sum of diagrams contributing to the self-energy of the field to be written as a geometric series, greatly simplifying the analysis. A momentum-independent solution to the self-energy is assumed. In the commutative case this solution is stable but nonrenormalizable. It is found that the noncommutative theory behaves similarly, but the nonrenormalizability is less severe. In Chapter 3 the fermionic Gross-Neveu theory is explored in the noncommutative regime.  T h i s theory is the fermionic analogue of the 0  4  vector model in that it has a  quartic interaction. Again there are two possible interactions but only the symmetric one is studied. T h e critical dimension is two.  Chapter 1.  3  Introduction  In two dimensions the Gross-Neveu model is marginally renormalizable by power counting.  T h e commutative version, however, is perturbatively renormalizable in the  coupling constant; this is also true in the noncommutative case. Here, though, the model is studied in a 1/N  expansion which is nonperturbative in the coupling since it sums  contributions to all orders in the coupling. Unlike in the commutative case, the theory is found to be nonrenormalizable in the sense that the dependence on the coupling cannot be removed. A running coupling is introduced, and this effective coupling runs to zero as the cutoff is taken to infinity, leaving behind the trivial free field theory.  T h i s is a  very interesting result, since the noncommutativity has destroyed the renormalizability despite early expectations that noncommutativity could actually improve it! Since the study is conducted in two dimensions, space and time are noncommutative. In this case the action contains infinite numbers of time derivatives, which ruins the ordinary Hamiltonian interpretation. B u t these are physical constraints and do not affect the mathematical analysis, so the study is justified on the basis that generic properties of the theories may still be exhibited. In Chapter 3 the Gross-Neveu model is also studied in three dimensions, where the coupling constant has the dimension of inverse mass. T h e commutative model is nonrenormalizable in the coupling constant expansion, but is renormalizable in the nonperturbative large-iV expansion at the second-order phase transition.  A s in the two-  dimensional case, the noncommutativity destroys the renormalizability of the large-./V expansion even near the critical point. Finally, in Chapter 4, the conclusions are summarized and discussed in the context of string theory.  Chapter 1.  1.2  4  Introduction  Quantum Field Theory  Q u a n t u m field theory is described by an action S expressed as a functional of a Lagrangian C through  (1.1) where D is the dimension of the space-time manifold. In this thesis the manifold will always be R  D  with Euclidean metric.  d^d) their space-time derivatives.  Here d> represents the field configurations and  T h e field configurations <j> which minimize the action  correspond to the observed fields. Using the calculus of variations the task of finding the fields which minimize the action of a particular Lagrangian can be reduced to finding fields which satisfy their Euler-Lagrange equations of motion  (1.2) Analogous to the partition function of statistical mechanics, quantum field theory has a generating functional  (1.3) T h i s integral is over all possible field configurations. A source term J has been included explicitly. T h i s allows correlation functions to be obtained from Z by taking functional derivatives with respect to J. T h e integral (1.3)  can be completed explicitly, for arbitrary space-time dimension,  when the integrand is a Gaussian.  A Gaussian integrand corresponds to a free-field  Lagrangian, which is quadratic in fields. In general, though, the path integral is not well defined and except in a few special cases must be treated through an approximation scheme.  Chapter 1. Introduction  1.2.1  5  Saddle-Point Approximation  In some cases this integral can be completed using the saddle-point approximation [8]. In mathematical contexts this is typically called the method of steepest descent, although in complex analysis it is also referred to as the stationary-phase method. In field theory it is sometimes referred to as a mean-field approximation. This technique requires that the action have a sharp m i n i m u m in field space.  The  minimum is found by solving the Euler-Lagrange equation of motion (1.2). A t the minimum, corresponding to the classical field configuration <f) , the integrand is maximized so 0  that the dominant contribution to the integral comes from this point. It is possible that the action has several minima; in this case the contributions from these minima should be summed. T h e validity of the saddle-point technique for the action uS is governed by the scale factor v.  For suppose the action has a single shallow minimum. T h e n when v is large  the integrand of the partition function is very large around the minimum of the action, and falls off quickly. In this case the saddle-point approximation is accurate even to low order. O n the other hand when v is small the corrections will be much more important. T h e action evaluated at  <f>o is the classical action, and corrections to the action are  quantum corrections proportional to 1/v.  In fact v is always proportional to 1/h so that  as h goes to zero classical field theory is obtained. In this document h will be set to one for convenience, so this limit will not be obvious. T h e terms in the perturbation expansions can be represented graphically by Feynman diagrams. These diagrams contain no new information but often provide a useful visualization tool. T h i s tool has led to the interpretation of higher-order corrections as corresponding to multiple self-interactions.  In this representation the n - o r d e r term in th  the expansion corresponds to graphs with n loops, so for example first-order corrections  Chapter 1.  Introduction  6  are often referred to as one-loop corrections. It is often possible to rewrite an action so that it has an explicit coefficient v. A s an example, consider the action of a scalar field with a polynomial interaction  S = <pD<p + \4> .  (1.4)  n  Here D is an appropriate differential operator. For an interaction of this sort one scales the fields by a power of the coupling constant A to make u scale as an inverse power of the coupling. So here rescale the fields as <fi — > A ^ 1  2 -  " ^ . Under this rescaling the action  becomes  S = X^  (if;Dtp + ip ) = X^S'.  (1.5)  n  -2  Now for this theory the expansion parameter would be v = X - » . T h e limitation to this 2  approach is that interesting features far from the small-coupling limit cannot be explored. A t the classical point d> , 0  ^= so that here — S[<j>o] + /  J(f>o  m  Jand  >0  (L6)  is a maximum. T h e action can now be Taylor-expanded  about the mean field d) . T h e linear term will not appear because of (1.6), so Q  oo  uS[4\ = uSfa] + ^ [</>o] 2)  Here  S^[(f)o]  refers the the n  t h  -  <f>of +  E %S< i<M (* n)  </>o) •  (1.7)  n  derivative of the action evaluated at </>. 0  In practical  applications the sum over n is often finite as higher derivatives of the action vanish. Now the field d> can be written as a classical field plus corrections, <fi — d> + v~i<j>. 0  T h e n the expression for the generating functional is given by  Z[J] =  j^  -" W -5 s  e  e  s ( a )  W* e-5:r=3 a  T ^ W * " .  (1.8)  Chapter 1.  As  Introduction  7  the first exponential is constant, it can be removed from the integrand.  T h e last  exponential can be Taylor-expanded in its argument as „  l-n/2  e - E - a ^  (  -  -I  OO  " W »  =  1  +  E  /  co  ,.l-n/2  \  J_ ( - Z ' — S ^ S " )  =l  m  m  -  V  n=3  U  -  m  •  (1-9)  )  T h e n to first order the integrand is Gaussian and the integral can be represented as the determinant of an operator,  Z[J]  = e~ ^ s  (S  det "5  ( 2 )  M) - • • •  (1-10)  where the omitted terms correspond to the perturbative corrections of order i / ( m  The  1 -  "/ ). 2  perturbation expansion expresses interactions in terms of free-particle correlation  functions. To  evaluate the determinant it can be rewritten as  detA = e  =e  lndetA  The  (1.11)  trace is now readily evaluated when In A can be diagonalized.  1.2.2 The  .  TrlnA  Effective Action  dependence of the generating functional (1.10) on the source term is hidden in the  definition of the classical field configuration because of  (1.6). In this way the source  generates a given classical field [9]. Given a fixed source, the free energy can be defined as  F[J] But  = - In Z[J] «  S[4> ] + 0  ^ T r l n (S [</>o]) •  (1.12)  (2)  this is still a functional of the external source, and it would be more natural to have  the energy as a functional of the classical field itself. T o do this, the free energy should be Legendre transformed with respect to the source. T h e result is the effective action  r[0 ] o  = F - jd xJ £. D  5  (1.13)  Chapter 1.  Introduction  8  T h i s effective action is extremized by the classical solution when the source is set to zero. T h e effective action is an extensive quantity; it can be written as an integral of an effective  Lagrangian over space-time volume.  Since the interesting classical fields to  be studied are often independent of space-time it is convenient to define the intensive effective potential  V  e/f  = Info]  [fa]  (1.14)  where v is a factor of space-time volume.  1.2.3  Large-iV  In theories where the fields satisfy an O(N)  symmetry, an expansion in N is possible and  is particularly suitable when N is large [10, 11]. T h i s approach can reveal information about the theory that is nonperturbative in the coupling. For example, a theory which is nonrenormalizable order-by-order in the coupling constant may in fact be renormalizable at a critical point.  B u t since the critical point may not be at small coupling a study  perturbative in the coupling may not apply. T h e existence of the O(N) be rewritten in terms of  symmetry allows the path integral over the fields </> to l  d> 4> . %  %  T h i s is analogous to a spherically symmetric integral, for  which the only nontrivial integral is the integral over the length coordinate. In this case the saddle-point approximation can be used with v — N. T h e typical model for this type of analysis is the bosonic O(N)  dfi theory, whose  action is  S\<\r\ = j  dx 4  where d> is <f> <j> summed over i = 1... 2  %  1  (1.15) N.  T h e large-AT limit is taken holding A fixed. Consider for example the propagator to one loop. If the vector index in the loop contracts with itself, the factor of N from the  Chapter 1.  9  Introduction  loop contraction will cancel the 1/TV of the vertex to give a contribution of zeroth order in TV. O n the other hand if the loop index is contracted with the external propagator the graph will go as 1/TV. A t two loop order there is again a contribution of first order, as well as contributions like 1/TV and  1/N . 2  In contrast to the small-coupling expansion, which considers all graphs order-byorder in the number of loops, to lowest order in N the large-TV expansion  considers  contributions from all loop orders and thus all powers of the coupling. However at each order in the coupling only some of the graphs are considered. T h e terms that are included in the expansion to first order are called bubble diagrams or daisy diagrams; they must have equal numbers of vertices and loops. Thus although this model is solvable in this approximation, it is nonperturbative in the coupling. This is the essence of the large-TV limit whenever it is used. To solve this model an auxiliary field cr is introduced through  (1.16) Since the model has an O(N)  symmetry the 0(TV)-direction of the fluctuations about  saddle-point will not affect the variation of the action. It is useful to choose the fluctuations about the saddle-point to lie in the 4> direction. T h e remaining (j) , i = 1 . . . TV — 1 N  1  are renamed ir\ i = 1... TV — 1. After the introduction of o the integral over the TV — 1 n  1  is Gaussian and can be  completed to give TV — 1 determinants. T h e partition function is then  Z = J V</> Vae- >>W  (1.17)  s  N  where here S'[o, 0"] = / dPx ^  (-d  2  + o) r  - ^ a  2  +  + \(N  - l)Trfn(-d  2  + o). (1.18)  Chapter 1.  Introduction  10  For space-time independent fields, the saddle-point equations are  5S 5<p  N  a<f>" = 0  (1.19)  and  5S_ 6a  1.2.4  i*V-£(*--»') i(*-i)/ +  dp  1  D  (2n)  D  p + o' 2  (1.20)  Renormalization  T h e integrals in these approximations always diverge. Fortunately these divergences can often be removed with a systematic physically sensible method. Theories where this is possible are said to be renormalizable, while all others are nonrenormalizable. In particle physics a major goal of the study of quantum field theory is to distinguish between the two classes of theories. T h e divergences are called U V divergences as they arise from integrands which diverge at large momenta. One standard approach to handling these divergences is to reduce the upper limit on the momentum of integration from infinity to a finite U V cutoff A . T h i s new parameter is unphysical in particle theory, so experimentally available parameters must be independent of it. T h e removal of the dependence on A involves the use of so-called running parameters in place of the original bare parameters. W h e n these parameters are forcibly introduced in a nonrenormalizable theory, the running coupling, which replaces the bare coupling and can be measured experimentally, flows to zero as the cutoff is taken to infinity. T h e resulting theory is simply a free field theory.  1.3  Moyal Product  Noncommutative geometries have a long history [12]. Their most recent appearance in particle physics is due to an idea from string theory that perhaps the space we live in is  Chapter 1.  Introduction  11  itself noncommutative. In the simplest noncommutative generalization of Euclidean space the position operators have a constant nonzero commutator given by  [x",x ] v  = -i6' .  W h e n the noncommutativity tensor 9^" vanishes, T h e effect of a non-zero invertible 9  liV  (1.21)  M/  becomes an element of ordinary R . D  on quantum field theories will be explored in this  thesis. W h e n quantum fields are written as expansions in these noncommuting coordinates the fields themselves become operator-valued. T h i s makes traditional methods very awkward to deal with. Fortunately, the algebra of this new class of theories is equivalent to the algebra of functions of real numbers when products between functions are replaced by the Moyal star product given by  f(x)*g(x)  1.3.1  = limexp [ ~ \ ^ f ( x ) g ( y ) .  (1.22)  Construction  T h i s correspondence can be constructed as follows [13].  Let d>(x) be an ordinary field  in R , and let <f)(k), a field in commutative momentum space, be its ordinary Fourier D  transform.  T h e n the Weyl operator or Weyl symbol associated with (j)(x) is given in  noncommutative position space by  m= / ( J ^ ^ y * * ;  (i-23)  this is just the analogous operational inverse Fourier transform of <f>(k). T h e n the mapping from 4>(x) to <p(x) is given directly by  4>{x) = [ d xcj)(x)A(x,x) D  (1.24)  Chapter  1.  Introduction  where the map A(x,x)  12  establishing the Weyl-Moyal correspondence is given by  \dete\J (27r)^  { X , X )  =  Note that when 9^  v  e  6  6  A(a;,£) .  (1.25)  t  vanishes the correspondence operator reduces to  A (a;, x) = 5(x — x)  (1.26)  as it should. A trace operator for the ip fields can now be defined by  Tr0 = j  d x(j)(x) D  (1.27)  which satisfies the normalization condition  Trh(x,x)  (1.28)  = 1.  W i t h this definition the correspondence operators are now orthonormal so that  Tr[A(x,x)A(y,x))  =8{x-y).  (1.29)  In addition the correspondence is now one-to-one since  4>(x) = T r  (J>(x)A(x,x)).  (1.30)  Derivatives of the fields <> \ living on the noncommutative space should also be defined. T h e important commutators [14] are  iS  (1.31)  and  4,#J  - -«'(0  (1.32)  Chapter  1.  Introduction  13  Note how (1.21) has forced an analogous deformation of the derivative algebra. T h e n  4 , A ( x , £ ) ] = -0,,A(x, z)  (1.33)  and = y d xdn4>(x)A(x, x)  [4,0]  (1.34)  D  follow. T h e translation generators are now given by the unitary operators e  wd  for v a  c-number, so that e -°A{x,  x)e~ -  iv  iv  B  = A{x + v,x).  (1.35)  Finally, consider the products between fields. Using the definition (1.24) above,  fa(x) = 4>i(x)fa(x) = J d yd z<j (y)4> {z)A{y, D  D  )l  2  x)A(z, x).  (1.36)  But, using the trace operator, <f> (x) is given by 3  4> {x) = 3  Tr^^Afx.z))  =  j d xd yd z(j) (y)(j) (z)A(y,x)A(z,x)A(x,x)  =  (  D  D  1  2  ^  )  2  ^  /  ^  =  1  ^  (  faWe-^Vfaix)  = fa +  where d  (1.37)  D  acts only on <p\ and d  fo  2  L  3  8  )  (1.39) (1.40)  on <j> . This establishes the equivalence of the algebra 2  of functions having ordinary products on noncommutative space with the algebra of functions having the Moyal product on commutative space.  Chapter 1.  1.3.2  Introduction  14  Properties  Although it is clear from (1.39) that the star product is not commutative since 9^  v  ^  9  Ufi 1  it is associative. T h i s follows directly from the associativity of ordinary multiplication:  (<kfa)  h  =  J  d C f c l (  ^ y  f c 3  )  ^(fci)e -*^(^)e'*»-^(fc,)e^-* t t l  = k  (fcfc)  (1.41)  Another very useful property is that under an integral one star product can always be dropped. Using (1.38) this becomes a straightforward calculation. Dropping explicit reference to the vector indices,  /  d  °  x  h * h  = (2ir)»\det9\  I^^^iCy)^^)^^" ^" ^*^" - ( ) 1  1  10  The first term in the exponential immediately vanishes by the antisymmetry of 9. T h e n completing the integral over x leaves 5(9' (y — z)). 1  Since this implies y = z, the final term  in the exponential vanishes and completing the integral over y cancels the normalizing prefactors to leave an ordinary product between the fields. A corollary of this is that products under integrals are invariant under cyclic permutations of the  fields.  These  properties will be used to simplify Lagrangians, which always appear as integrands.  1.4  Noncommutative Quantum Field Theories  Using the Moyal star-product and its properties the methods of ordinary quantum field theory can be used to study the noncommutative theories.  Because of the Weyl corre-  spondence there is no need to deal with operator-valued functions. Instead, to construct a noncommutative quantum field theory, use the Weyl correspondence to replaces the products between ordinary commutative fields by the Moyal star-product. But  by the discussion above the star-products have no effect on the terms in the  Lagrangian that are quadratic in fields. However, this does not mean that the noncommutativity has no effect on free field theories since the star-product will still appear in  L42  Chapter 1.  Introduction  operator product expansions for example.  15  T h e noncommutativity really does do more  than just change the form of interactions. Since the nonlocality of the Moyal product distinguishes between noncyclic permutations of momenta about vertices, an important new distinction between planar and nonplanar diagrams in momentum space arises. Diagrams which are planar in momenta are modified from ordinary Feynman diagrams only by constant phase factors, that is, phase factors that depend only on external momenta and are independent of the loop momenta. O n the other hand, phase factors appearing in nonplanar diagrams depend on the momenta of integration and thus affect the behaviour of the integrals [15]. Although the phase factors in the nonplanar diagrams can improve convergence of integrals over loop momenta, this need not be the case [16, 17].  B u t even when the  noncommutativity softens the ultraviolet ( U V ) divergences, which can still be removed through a usual renormalization prescription, new infrared (IR) divergences crop up which can destroy the renormalizability. T h i s is because after completing the nonplanar integrals with the cutoff A , the result depends on A through a new effective cutoff  A ff = e  1 /2  (A  - 1  + p99p)~ ' . For nonzero external momenta the effective cutoff approaches a finite  limit as A is taken to infinity. Thus these nonplanar diagrams are no longer UV-divergent. However, once A has been removed the effective coupling is IR-divergent. T h i s the U V / I R mixing of [18, 19].  Chapter 2  Noncommutative O(N) 0 Theory 4  2.1  Introduction  One simple theory to study is the O(N)  scalar (f) theory in the large-N limit. In the 4  commutative case a local auxiliary field can be introduced which allows the resulting path integral to be calculated by a saddle-point or mean-field approximation. T h e expansion parameter is 1/N.  T h i s solution is exact in the sense that it sums contributions from all  orders in the coupling. T h e commutative 0(iV)-symrnetric c/> theory is described by the Euclidean Lagrangian 4  £ = I^(-a + m ) ^ + A( y) . a  a  2  0  (  2.i)  Here i is a group index running from 1 to iV and Einstein's summation notation applies. T h e (f> are N bosonic fields invariant under 0(N) %  coupling are given by m  2  transformations. T h e bare mass and  and A respectively.  In four dimensions A is dimensionless and the theory is marginally renormalizable. In the large-TV limit, for A > 0, this is a free theory unless a U V cutoff is imposed [20]. T h e n the ground state of the theory depends on the values of the cutoff and bare mass and coupling. In the large-TV limit it is determined by the sign of  m  A  2  T  +  2  32^  <"> 2  2  where A is the U V cutoff. T h e ground state exhibits spontaneous symmetry breaking to an 0(N  — 1) state if (2.2) is negative, but otherwise maintains the full O(N) 16  symmetry.  Chapter 2. Noncommutative  0(N)  tf  Theory  17  O f course this analysis is valid only for momenta much less than A . W h e n A < 0 the theory is inherently unstable. The  noncommutative version of this theory has two interactions which are distin-  guished by the cyclic order of the momenta about the vertex;  C = \tf (-d + m ) tf + ^ 2  (tf * tf) (tf * tf) + ~tf *tf*tf*  2  tf-  (2.3)  W h e n 9 goes to zero the star-products become ordinary products and the coupling is given by A = | (Ai + A ). It is useful to rewrite the interaction portion of the Lagrangian 2  in momentum space, where it becomes Ant = /  (ft  |^)  e  ^  ^  M  ^  D  ^  D  )  {  tf iPi) tf ( f t ) tf ( f t ) tf (P4)  p  i  +  W  p  2  +  p  3  +  p  i  )  + ^fS***}  X  •  (-) 2  4  Although less obvious in momentum space, the A i vertex is symmetric in the fields, whereas the A vertex is not.  T h i s will be established concretely in the next section.  2  In Section 2.3 the propagator is considered for large-AT and small coupling. T h i s allows the full theory, with A nonvanishing, to be studied. However summing contributions to 2  all orders in the coupling is not possible.  In an attempt to find other nonperturbative  solutions to the theory with A present, the Schwinger-Dyson equation of the theory is 2  introduced in Section 2.4. A recursive definition of the self-energy is found, but nonperturbative solutions dependent on A are not discovered. 2  The theory is then studied with only A i present. T h e theory then becomes very similar to the commutative one.  T h e traditional method of solving the model will work, so in  Section 2.5 the saddle-point approximation is introduced explicitly. T h i s approximation has the effect of summing contributions to the propagator from all orders in the coupling Ai . T o lowest order the theory is identical to the commutative model, so the same phase structure is found. Q u a n t u m fluctuations, which do depend on the noncommutativity, are  Chapter 2. Noncommutative  0(N)  18  <fr Theory 4  studied in Section 2.6. Compared to the commutative fluctuations, the noncommutative corrections are found to reduce the stability and improve renormalizability.  2.2  Local Auxiliary Field  T h e usual method of solving the O(N)  model in the large-N limit involves the introduc-  tion of a local auxiliary field o ~ 4> (j) . W h e n this field is introduced the Lagrangian is l  1  quadratic in both the 4> and o. Completing the path integral over u results in the origil  nal Lagrangian, while completing the path integral over the <f> leaves a new Lagrangian 1  which depends only on o. It is this new Lagrangian which can be studied in the large-N limit using the saddle-point approximation. B u t for this method to work it is important that the original Lagrangian have an interaction symmetric in $ (\>. %  l  In the present case, to determine whether o can be successfully introduced only the exponential term in (2.4) must be checked. This is because the Kronecker delta functions can clearly not be written in any other way. O f course the momenta of integration can easily be relabeled if necessary. Denoting p^O^p ^  by p  1  m  x p, n  the exponential is  g f [Pl (P2+P3+P4)+P2X(p +P4)+P3Xp4]  ^2  X  3  5)  T h i s can be rewritten using conservation of momentum and the antisymmetry of the product as g£(PlXP2+P3Xp )  (2.6)  4  or as e  ^(PlXp3+P2Xp4)gip2Xp  (2  3  7)  from which it is clear that the A i term factors symmetrically while the A term does not. 2  Thus no local momentum-conserving auxiliary field exists. T h i s is an unfortunate result, but it also reveals the deep impact of the noncommutativity.  Chapter 2. Noncommutative 0(N) </> Theory 4  2.3  19  Two-point Propagator  T h e propagator, or two-point correlation function, gives the amplitude for a particle to propagate from one point in space to another.  For an interacting theory, such as the  one studied here, this amplitude is modified from the free-particle amplitude by selfinteractions. In the large-iV limit, ordinary perturbation theory in the coupling can still be used. In momentum space the propagator has a single one-loop correction in the large-iV commutative model. Here, since there are (|) cyclically distinct ways to order the momenta about the single vertex, there are six one-loop corrections to the propagator. the diagrams that are not suppressed by 1/N  only the O(N)  T o find  index structure needs to  be considered. T w o important contributions are found; the correction to A i is planar in momentum space while the other is a nonplanar correction associated with X  . In light  2  of this it is not surprising that it is the A vertex which prohibits the existence of a local 2  auxiliary field. Using the result above it is now straightforward to calculate the inverse two-point propagator r = S^  to one loop order when the couplings are small and N is large. T h e  1  planar correction is Ai f d k 1 CP _ _ L / — 8 J (2TT) k + m 4  c  A  v  1  4  2  x  Tr  2  = 2  8(2TT)  (  k2  1  4  2  A - m \  / A In — \m l  2  2  \  _ \ + constant terms  J  , . (2.8)  x  n  o  )  and the nonplanar correction is  T  C.«P 2  =  * L /• 8  J  ******  (2?r) k 4  2  =  + m  2  (A  - m In  2  8(2?r)  4  \  e  2  f  f  (%U \m ) 2  constant terms) ,  (2.9)  )  where 1  .  .  Chapter 2. Noncommutative  0(N)  tf  Theory  20  and A is a U V cutoff. Thus the inverse propagator to one loop,  , is given by [18] as  ^='+™+( - ™>°(£))+^ K - ™ (^)) • 2  A2  2  2in  Note that as p only occurs in the combination p9 in the one-loop corrections, this theory will look planar in the small-momentum limit, at least naively.  However, this  limit needs to be taken carefully, and is discussed in [18]. One can certainly add multi-loop corrections to this result; this corresponds to higherorder corrections in ordinary perturbation theory.  Ideally one could add contributions  from all orders in the coupling immediately. However as described above this will not be possible when A is nonvanishing. W h e n A does vanish the resulting expansion in A i is 2  2  a geometric series, whose sum is of course known. In fact the series, which is defined as the self-energy, corresponds precisely to the auxiliary field which will be introduced in Section 2.5.  2.4  Self-Energy  T h e self-energy E of a particle is the sum of loop corrections to the bare propagator defined through s  = VTs  <'> 21 2  so that m? + E is the effective mass of the particle. Here S is the full propagator and SQ  1  — (p + m ) 2  2  is the inverse bare propagator.  In the large-Af limit of commutative tf theory the only diagrams which contribute to E are the bubble or daisy diagrams, which have equal numbers of loops and vertices. These diagrams survive the large-N limit since each loop provides a factor of N and each vertex goes as 1/N.  T h i s sum of bubble diagrams can then be factorized into a geometric  series and the sum can be computed.  Chapter  2.  Noncommutative  0(N)  (j)  21  Theory  4  It has been shown in Sec. (2.2) that a local auxiliary field cannot in general be introduced into the noncommutative version of this theory.  Equivalently, the bubble  diagrams can no longer be factorized into a geometric series. A  2  T h e problem with the  vertex is that the phases entangle adjacent loops so that a two-loop contribution  is not the square of the one-loop contribution. Thus, the self-energy must be studied directly through its definition (2.12). T h e full propagator is S (q) =  8  mn  S (q)6™  +  0  =  (<F  (q)  <t>  n  {-q) C  ( , 2,PZ, A))  int  Pi P  P  S (q)6  mn  0  +  I ^ ^ e ^ < ^ H ^ )  X 46 S (q)6(q mi  0  - ) Pl  D  i  W  {8 V S(- )8 kl  (-p  n  P3  3  + P 3 + P 4 ) (^M" - p) 6 (~ 4  + V 5 S(- )5  (-P2 - )  + 8 8 S(- )8  (-P2 - p ) 8 ( - P 4 - q)} S(q)  l  kn  P2  jk  Pi  ln  P2  0  3  +  |^*)  - q)  P2  - q) (2.13)  3  which to first order in 1 /N S{q) = S (q)  8 (-p  (Pi  + /  after summing over the group indices leaves  T ^ p ^S (q)S(p)S(q)  + ±-J>*<S (q)S(p)S(q)}  0  In the large-iV limit this gives the self-energy £ = S '  -  _ [ ^ P _  J (2TT)  d  .  0  1  + m? + E(p)  - 1  — SQ  (^_  V 2  as  1  +  (2.14)  ^  e  2  i  P  x  q  )  (  )'  2  1  1  5  )  ]  If the general solution to this equation could be found, it would allow a very detailed nonperturbative study of the theory. B u t since it is not clear how to solve this equation in general, solutions must be found perturbatively or in special limits of the theory.  Chapter 2. Noncommutative  0(N)  4>  A  22  Theory  T h e nonplanarity of this theory is associated with A , so to simplify (2.15) without 2  losing interesting behaviour it seems best to set A i to zero.  Interestingly, unlike the  commutative case, there is no momentum-independent solution. However, perturbative solutions can be found. Gubser and Sondhi [21] argue that this leads to a striped phase due to the oscillatory phase associated with A . T h e contribution of the A term at large loop momentum 2  2  diverges as the external momentum goes to zero. This prevents the condensation of the small momentum modes of the fields. Therefore the ordered state to which the fields condense must break translation invariance. T h i s will not be studied here. O n the other hand, setting A to zero allows only the momentum-independent solution 2  of the more familiar commutative theory. Noncommutative corrections will still be found when perturbations about the solution are considered. T h i s will be studied below.  2.5  Symmetric Vertex  Keeping only A i in the theory allows progress to be made. T h e terms in the bubble series now factor and the local auxiliary field can be introduced. T h i s allows the running of the coupling and the phase structure of the theory to be explored. A t the classical level all dependence on 9 will disappear, but quantum corrections will depend on the noncommutativity. T h e noncommutative quantum corrections will mildly affect the stability of the phases and the renormalizability. T h e Euclidean Lagrangian is given by  CW} = l<t> (~d + m {  2  2  )  P +^  (f  * ^) . 2  (2.16)  T h e interaction now factors, so a local auxiliary field can be introduced. Note that the  Chapter 2. Noncommutative  0(N)  </> Theory  23  4  auxiliary field o does not have a kinetic term. T h e resulting Lagrangian is  TV  (2.17)  2Ai  Here and below, the notation -kio-k indicates that the •-product is important in the relevant operator expansion. Although this Lagrangian has an extra albeit nondynamical field, it describes the same physics as the original Lagrangian.  T o see this, one must  consider the path integral of the exponentiated action. T h i s is given by  (2.18)  T h e original Lagrangian is obtained by performing the path integral over cr. Alternatively, one may integrate over the (j) to leave a Lagrangian which depends only on cr. T h e 1  resulting description will be solvable in the large-TV limit. Before integrating over the <f) a shifted auxiliary defined by u 1  2  = m  2  + io will be  introduced. T h i s simplifies the expressions for the propagator, although it introduces a tadpole. Now, letting J be the source term for the (f>\ the resulting generating functional 1  of correlation functions is  Z[f]  =  / VtfVp  2  exp (2.19)  Since p  2  does not correspond directly to an observable field it is not given a source term.  If it were given a source term, then the TV + 1 degrees of freedom in the sources would not be independent. It is convenient to introduce an analogue of the Helmholtz free energy as  \nZ[J%  .  (2.20)  Chapter 2. Noncommutative  0(N)  tf  Theory  24  since then  jjr = - M , - * -  <>  s  where <f>  l 0  2 2 1  are the classical solutions. Now the generating functional of one-particle irre-  ducible diagrams is  r [ ^ , /x ] = F - J d xJ D  2  l  6F  (2.22)  5^  so that \Ji—n =0 —  (2.23)  U>=o — 0.  l  Hi  Before proceeding further, the integral over (jf in Z[J ] will be completed. First, shift l  1 -d  + *[x *  d x \tf  ( - d  2  and note that Vtf  Z[f}  =  = Vtf.  (2.24)  2  Now  JVtfVn exp{-  j  2  D  2  + ^ ) t f - ^ - ^  2  - m  2  y  L  -Iji  2 -d +V*' 2 jP/i exp|-yTrln (-d + V*) + 2  2  2 -o + *p * 2  2  Jd x D  N 2A7 (2.25)  T h i s latter integral can only be evaluated when iV is large, since then the saddle-point approximation, or method of steepest descents, is valid. T h i s is the reason that the earlier analysis focused on the large-iV limit. T h e argument of the exponential in (2.25) must be minimized as a functional of u for 2  fixed sources. T h i s is how u  2  acquires a classical interpretation. Introducing the classical  //Q as the m i n i m u m of the functional in this way is equivalent to including a source term H for p  2  and subsequently solving the constraint placed on the N + l source terms J , H. 1  Since the classical solution is really the weighted average of all possible quantum fields,  Chapter 2. Noncommutative  0(N) 0  4  Theory  25  pi is also known as the mean field, and in this context the saddle-point approximation is often called the mean-field approximation. In mathematical discussions this technique is also referred to as the method of steepest descents. To lowest order this approximation yields  F[f}  = - y T r l n (-<3 + *// *) + J 2  2  N  dx D  1  (pi - m ) 2  2A7  -d  2  J  1  + *$*  2  (2.26) Finally, taking the Legendre transform  N I M * ) ] = j TV In (-d  2  + */. *) - f d 2  D  (pi - m )  x  2  -  2  2  \j  J + *tj%* 1  -d  2  (2.27) Using the definition of the classical solutions <p this can be rewritten as l  0  r[p (x), ft(a)] = jTr In (-d 2  2  dx  +  D  (p - m ) 2  2  0  2  - | f l (-d  2  +  4 (2.28)  For constant pi the star-products can be dropped and the T r l n can be evaluated, leading to  I K , <&{x)] = j J d  D  *lj05^{p  2  + ti) ~ 2 ^ " {ti - ™ f + ^  {~d + ti) 4 2  (2.29) It is also convenient to introduce the effective potential  V  = lr[p ] = ^ T r In (-d 2  eff  2  0  + p) - ~ 2  (/xg - m ) 2  2  +  (2-30)  which is minimized by the constant solutions p\ and (f> . Here v is an infinite factor of 0  space-time volume.  Since constant classical solutions have been chosen, the •-product  no longer affects the expansion to first order.  Chapter 2. Noncommutative  2.5.1  Four  0(N)  <p Theory  26  A  Dimensions  Now  'lf  -rf«..2,i  dV  ^  = 0  (2.31)  so that either (j) or u-l vanish. This means that the theory has two phases, and along l  0  the critical line separating the phases both 4>\ and / i , vanish. W h e n (f) is nonvanishing, 2  l  0  so that these fields have acquired a vacuum expectation value, the O(N)  — 1) symmetry. In the other  the original Lagrangian is spontaneously broken to an 0(N phase the 4>  % 0  symmetry of  vanish and the theory maintains the full symmetry.  In the unbroken symmetry phase where pi is non-zero, the mass-gap equation is  dVeff dp  _  N f  d*p  2 J  2  1  (27T)* 2 P  N  a  r 32TT2  N ^2  +  N  2  {*>  i  rn )  A N Nm 72 - T-fi + - T - = /Z§ AI ™ ' A 2  l  ln  2  0  ,  2  2  2  - ^rr^l 327r ^  A  A  n  n  n  .  n  ( -  0  2  3 2  )  X  in which a U V cutoff A has been imposed and it is assumed that pi <§C A . Thus in this 2  phase pi must satisfy 2  /  1  1  ^ { ^  +  ,  A  m  \  2  ^ 7 J  A  2  2  x r ^ So here the running coupling and mass should be defined as 1  1  A (M )  A  2  r  =  1  (2  , A + ^ l n — 32TT M  33)  . (2.34)  2  2  x  -  +  2  and m (M ) _ m 2  2  A  2  r  A (M ) 2  r  where M  2  is the renormalization scale.  ~~ Ax  +  2  32TT2-  Note that since A i is positive, A  {  r  Z  -  6  b  )  —>• 0 as the  cut-off is taken to infinity; this is the Landau pole. Using these definitions the mass-gap equation becomes simply  1*1 = m  2 r  (2.36)  Chapter 2. Noncommutative  0(N)  <p Theory  27  4  so that pi is identified with the renormalized mass of the (f> and acts as an order parameter 1  of the theory.  Note though that Bardeen and Moshe [20] choose pl/X  r  as their order  parameter. In the other phase the symmetry is spontaneously broken, pi = 0, and the mass-gap equation becomes dp  2  32TT  Ai  2  0  so that «  ml  = —r~-  (2-38)  Although in general all couplings up to the dimension of A and compatible with the x  symmetry of the A i vertex are needed to remove U V divergences, it was consistent in the large-iV limit to consider the case where A vanished. T h i s is because to first order 2  in 1/N  the U V divergences arise solely in planar diagrams where the ordering of the  momenta about the vertices is unimportant. B u t since either coupling can be associated with planar diagrams, the divergences can be removed by counterterms of either form. T h e nonplanar sector is UV-finite in the large-iV limit since the terms become convergent due to the phases. However, the nonplanar sector is plagued by I R divergences which can destroy renormalizability even when all U V divergences have been appropriately removed from the theory. T h i s will be studied below.  2.6  Beyond the Classical Case  In the above first-order phase structure analysis, 6 plays no role, so that the theory has reduced to the classical case. T o see the effects of nonzero 6, perturbations about pi, the minimum of T (2.28), must be considered. T h i s corresponds to including the first-order corrections to the saddle-point approximation in (2.26). It will be useful to reserve the symbol <fi (x) for the solution above, so that as an arbitrary <p  %  0  approaches <p , the field l  Q  Chapter 2. Noncommutative  0(N) <f> Theory 4  28  pi approaches zero. To properly define the expansion of  Trln(—d + pi +*i8o*), 2  So is  where  a small  perturbation about the classical solution pi, it is best to return to (2.25). Thus recall that  exp j - y T r l n {-d + p + *i6o*J j 2  2  0  = j Vft exp j-^ j d xft * (-a + a + iSo) *ftJ 4  =  2  2  y p ^ e x p j - ^ l d xft*(-d + p )*ft}f^^[-^J 4  d x5oft*4>^) (2.39)  2  2  Dividing out the infinite constant Z[J — 0], the n l  t h  4  n  Feynman diagram in this expan-  sion is  i\  n  I" (  _  1  n! / * ' ' ' ^ " M ^ O M ^ a ) • • • So(x )r (x ..., x )  2~)  d  X l  n  ly  (2.40)  n  with  T  [x\, . . . ,  X) n  \x-i) •ftixt)* • • • * 4> (x ) • 4> (x )) j  j  n  / IWOn) •  n  ji=Q  • • • ^'(*n) * ^ n ) e x p { - | / / Vfi exp { - § / d a;<^ ( - d 4  2  d xft (-d + p ) ft) 4  2  2  .(2.41)  + ^i)  Note that in the exponentials both ^-products could be dropped since pi is assumed to be constant, and the expression occurs in an integrand. Now define r to be the connected c  diagrams defined by r . T h e n the sum above can be rewritten as an exponential so that taking logarithms gives  iSo \  TV" , / -Trln 1 + 2  V  ~d + ti T NV °° (—i) r - — m=l V / d x • • • d x 8o(x ) • • • 8o(x )T (x , ...,x ) 2 ~~, m J N m=l 2  g (-0  m  4  4  x  c  m  1  m  1  m  i J d xiSo(x )T°(xi) + ^ j d xid x 5o(xi)Sa(x2)T (xi, x ) + ... 4  4  1  4  c  2  2  .(2.42)  Chapter  2. Noncommutative  With p  2  0(N)  tf  29  Theory  — pi + i8u the effective action becomes  lV,0i  fTrln(-a  =  - N  j  + ^) + | T  2  d x  2A  _  <2  - m ) + iSo) 2  X  1 + pl) p - ^ p +  tf (-d  2  2N  ((pi  A  iSo  l n ( l +  t  iS**?  Since 8 a is an auxiliary field, its equation of motion is simply  (2.43)  = 0. T o first order  in 8a, _8T_ 8(8a)  N  (x) + -J  :  d*x 8a(x )T (x,  x) --^-(p  c  1  1  x  -  2 0  m ) + —So + ~tf 2  *tf  = 0.  (2.44) B y conservation of momentum one may write T (X, XI) = T%(—id)8(x — xi). c  M*) ( \l2  •^tf-ktf  +  +  -r (x)  l  c  m - m) .  Thus  2 >  (p  2  2  0  (2.45)  Because pi still satisfies (2.32), the last two terms cancel and the expression reduces to i8a(x) -  " -(j)•*<f) l  2N  1  (2.46)  where here (2.47) *eff  ^1  *  So from (2.43), and using (2.32) again,  iWo]  =  jTrln  +J  (-d  +  2  d x \\tf 4  Note that the part dependent on  (-d  p  2  0  2  ) - £ - J d  + pi)  +  4  x ( p  2 0  (ti  -  2 m  )  2  * ti)'  (2.48)  has the same form as the original Lagrangian  (2.16), with the bare coupling Xi replaced by a new effective coupling A / / ; this confirms e  that the choice of X ff e  is appropriate.  Chapter  2.  But X ff e  Noncommutative  0(N)  (p  30  Theory  4  is important for another reason. For consider the second-order term in 5o  in (2.43); this is given by 1 [ d x5o^—5o.  (2.49)  4  2 J So X ff e  is the propagator of the o  A ff e  field.  If X fj  becomes negative the propagator is  e  tachyonic, meaning that the ti condensate, and the 0(iV)-symmetric phase, is unstable. T h e critical line (f) pl = 0 then corresponds at least to a local minimum. Whether this 0  is a global m i n i m u m is not examined here. To study the behaviour of X ff,  then, the four-point coupling  e  must be found ex-  plicitly. In Fourier space, dq  1+  4  T2  f \  1  f  d  1  W  +  [Q  2  e  iqxp  ti] [{q - P? +  ti  ^/o ^^^ {-24^ l  2TT  2  xp  11^ i n A  1  ^+ 2  (l  if)'P  i n  ln^ +  ( ^)} s  ^  2 < <  l n ^ ) , p  i)+M2o  (2  -  so)  (2.51)  » ^ ,  2  with 1 -r^+p9 p ~ A  (2.52)  2  A  2  2  eff  as before. Thus here  1  Ac//(p)  =  7 -  1  Ax  j r  /  1  +  I  +  N  ^ (p) 2 2  1  647T  ^  2  l  ln  n  A  2  p  2  1  1  64-K  2  ^ - e h  ln  l  A g 2 / /  p  2  a  (  1  +  •  p  2  »  *  -  (2 53)  Chapter 2. Noncommutative  0(N)  <fi Theory  31  4  B y taking a derivative of the exact expression (2.50) for  X  eff  it can be seen that the  effective coupling is a monotonically increasing function of momentum. It is always positive, which establishes the stability of the constant solution as at least a local minimum. T h e interesting difference between the above \ ff e  and the usual effective coupling of  the commutative theory is the additional logarithmic term in (2.53) which reduces the inverse effective coupling as the cutoff is taken to infinity. Thus the new effective coupling decreases more slowly than the commutative one for large cutoff. It also means that the solution is slightly less stable. T o summarize, the introduction of noncommutativity in the A i vertex has reduced but not destroyed the stability of the constant vacuum, and it has softened but not cured the nonrenormalizability. Gubser and Sondhi [21] suggest that the A vertex has further2  reaching consequences in that it completely destroys the stability of the constant vacuum and leads instead to a striped phase. T h e y find a first-order transition from the ((f)) = 0 phase to the striped broken-symmetry ordered phase.  Chapter 3  Noncommutative Gross-Neveu Theory  3.1  Introduction  This fermionic counterpart [22, 23] to the O(N) tf theory is more closely related to the Yang-Mills theory in that it is asymptotically free. T h e full Euclidean Lagrangian is  C[ft = -ft 0ft - ±  (ft* ftf - ±  (ft * ft *ft* ft) .  (3.1)  Here the spinors tp are taken to be Majorana fermions so that nonplanar diagrams appear l  in the leading order in the 1/N expansion. Being Majorana, they are symmetric under simultaneous charge and complex conjugation. In two dimensions they must have two components, while in three dimensions they can have two or four components. T h e y will always be treated as two-component spinors here. A s the arguments which prevented the introduction of a local auxiliary field in the presence of the X coupling apply here, in this analysis of the Gross-Neveu model A will 2  be set to zero immediately. Because of chirality, in the free fermionic theory the two components of the spinors have an O(N) symmetry separately. T h e Gross-Neveu A interaction breaks this O(N) x O(N)  symmetry to leave only an O(N) x Z symmetry. T h e remaining O(N) symmetry 2  is a diagonal subgroup of the original symmetry, and the Z  2  symmetry is the chiral  symmetry. A mass term was not included in the above Lagrangian since the chiral Z  2  symmetry forbids this. T h e fermions can acquire a mass only if the chiral symmetry is spontaneously broken. T h e order parameter for this symmetry breaking is (^ftft^.  32  Chapter 3. Noncommutative  3.2  Gross-Neveu  Theory  33  Local Auxiliary Field  A s found for the bosonic model after A was set to zero, a local auxiliary field can be 2  introduced. T h e resulting Lagrangian is  C[ft,a}  =  -ft(0  2N  *cr*)ft  +  + — a  .  2  (3.2)  A  Integrating out ft in the saddle-point approximation, as done for the bosonic model, gives the generating functional  Z[H] = J D a e x p { y T r l n (0 + *o*) - j  dPx  —a  -ft (d  2  0  +  ft  + Ha  } , (3-3)  and then the effective action is  T[ft,  a] = - y T r l n (0 +  + J dx D  ^cr  2  + ft (0 +  ft  .  (3.4)  Taking the fields to be constant, the effective potential becomes  Veff = ^Tr\n{0  + o) + ™o  2  +  ftfto.  (3.5)  As in the bosonic model this theory has two phases corresponding to o = 0 and ft  0  = 0 = ft , and the mass-gap equation is 0  dV  Nd  r dp D  1  AN  -•  ; i  ,  n  D  .  Here d = 2 is the dimension of the gamma matrices and v is again the volume of spacetime.  3.3  Perturbation About the Classical Solution  Since the effective potential depends on constant fields, the effects of noncommutativity can only be probed by considering perturbations about these solutions.  Chapter 3. Noncommutative  Let o — M  2  Gross-Neveu  + So where M  Theory  34  satisfies the conditions above, and So is a perturbation  2  about this classical solution. Now, using the formalism developed in the previous chapter, the effective action is given by r[VM  =  -yTrln(>+ M ) 2  -y  |y d xiSo(x )T (xi) D  - ^  c  X  + / dPx \ ^ M J LA  4  + ^-M So  d xid X SO(XI)SO(X )T (X , D  D  + ^5o  2  A  C  2  A  2  2  + $ U + M  1  2  ^  X) + ... 2  + So) ^1  J  •  (3.7)  In this chapter the r functions will always be the fermionic counterparts to the r functions described in (2.41). T h e formal definition is easy to obtain from (2.41) simply by replacing the boson fields 4> with the spinor fields ip and replacing the Klein-Gordon l  l  propagator d + *pl* with the Dirac propagator ^ + *<7*. T o calculate the commutative 2  r functions the parameter 6 can be set to zero immediately. Using the mass-gap equation for nonzero M  ST  5(So)  N ~2  J d x r (x, D  c  1  2  x )So(x ) 1  l  , the equation of motion for So is  AN.  + ^-So  + i>i^l = 0  (3.8)  Thus (3.9) with  T — = T + U(-id). A ff e  A  (3.10)  O  Substituting this back into the effective action gives a new effective action for the classical fields:  Wo)  =j  dx D  $(0 + *)^  hll fJ.SN  (3.11)  This analysis is completely analogous to that performed in Sections (2.5) and (2.6). To complete the analysis the dimension D must be fixed so that the integrals can be completed explicitly.  Chapter 3. Noncommutative  3.4  Gross-Neveu  Theory  35  Two Dimensions  I n t w o d i m e n s i o n s t h e effective p o t e n t i a l is  T h e classical value of M minimizes this p o t e n t i a l a n d corresponds t o t h e d y n a m i c a l l y g e n e r a t e d f e r m i o n mass.  S i n c e i t is a f u n c t i o n o f M  2  , t h e sign o f M is undetermined;  t h i s is d u e t o t h e c h i r a l i t y o f t h e m o d e l . T h e g a p e q u a t i o n is s o l v e d w h e n  \ = b»w-  < > 313  A f t e r t h e r u n n i n g c o u p l i n g is c a l c u l a t e d , i t w i l l b e p o s s i b l e t o r e p l a c e t h e d e p e n d e n c e o n A a n d A w i t h a d e p e n d e n c e o n M. T h i s t e c h n i q u e is c a l l e d d i m e n s i o n a l t r a n s m u t a t i o n . N o t e t h a t i f a r e n o r m a l i z e d c o u p l i n g i s d e f i n e d a t t h e r e n o r m a l i z a t i o n scale \x b y  A  r  A  47T  /J,  2  t h e n i f A i s t a k e n t o i n f i n i t y a l o n g f i x e d A , A m u s t go t o zero. T h i s i s t h e a s y m p t o t i c r  f r e e d o m o f t h e G r o s s - N e v e u m o d e l . T h e u n p h y s i c a l I R L a n d a u p o l e is also present here; t h e r e n o r m a l i z e d c o u p l i n g goes t o i n f i n i t y as /J? is t a k e n t o zero a l o n g f i x e d A a n d A .  3.4.1  The Commutative Theory  To study the four-point coupling A the four-point correlator must be found.  In the  c o m m u t a t i v e case t h i s is g i v e n b y 1 T  C 0  =  - -  2TT  M  ^  \ V  4 M  2  2M  (3.15)  E u l e r ' s c o n s t a n t is d e n o t e d b y 7. T h e r e g u l a r i z a t i o n u s e d here t o t a m e t h e U V divergences is t h e s a m e as t h a t u s e d i n [18, 19]. U s i n g t h e g a p e q u a t i o n t o e l i m i n a t e A, t h e effective  Chapter 3. Noncommutative  Gross-Neveu  Theory  36  coupling is  K„  = ~j=  — ^ -  ^m(yrrg: i) +  2M  . +  7  (3.16)  -i  For p 3> M , the coupling 2  2  Xeff «  (3.17) 1 1 1  M  becomes small, as expected. A s the momentum is lowered to small momenta the coupling increases, but it stops increasing when p reaches the scale M, at which point the coupling freezes at eff  X  3.4.2  «  (3.18)  r.  7 —1  The Noncommutative Theory  A s in the bosonic model, Tj receives contributions from both planar and nonplanar graphs through r | =  + T%  NV  when 9 is nonzero. T h e planar contribution T^ is again half of 9  the commutative four-point function. O n the other hand, the nonplanar contribution to the connected four-point function is  q9 2  ~4~  where K  0  _1_  2  +  A*  +  denotes the modified Bessel function. Note that this term has a finite limit as  A is taken to infinity, unlike the planar contribution. If A is removed from the resulting expression for X ff e  the dependence on A is not removed so that for any p >  the  effective coupling vanishes as A is taken to infinity. T h e U V behaviour of A / / can be studied in the limit where both p  2  e  M  2  < A . 2  <C A and 2  Chapter 3. Noncommutative  T h e n when p  2  3> AM  2  Gross-Neveu  and p  »  2  Theory  37  the planar contribution is  r ^ - ^ l n ^  (3-20)  p  z  8-7T  while c,np  T  _  -M8p  (3  e  2 1  )  so that  * *-2ln£ - d i g '  A e / /  ( 3  -  2 2 )  T h e bare coupling A has been removed using the gap equation. But when p  2  <  p  2  < AM ,  and p >  2  ^  2  then  ^-^•4  <- > 3 23  and r ' c  n p  2  «  In  (e p M ) 2  2  2  ;  (3.24)  thus, using the gap equation again,  Here the effective coupling still goes to zero as the cutoff is removed for finite nonzero external momentum. Therefore in this noncommutative theory renormalization does not remove the cutoff dependence. T h e interaction goes to zero as the cutoff is taken to infinity, leaving behind the trivial free field theory. T h e arguments above depend heavily on the satisfaction of the gap equation. It is certainly possible to choose the bare coupling so that it does not satisfy the gap equation but rather removes the U V cutoff dependence. the model.  T h i s approach destroys the stability of  Chapter 3. Noncommutative  3.4.3  Gross-Neveu  38  Theory  A Double-Scaling Limit  A double-scaling limit can be taken by sending A —> oo and 9 —>• 0 along fixed 9A = jfc for some fixed parameter C.  This corresponds to a regularization of the commutative  model since the theory is noncommutative only at energy scales beyond the U V cutoff A . Even though the noncommutativity lives beyond the U V cutoff, it still affects the theory at all scales through the U V / I R mixing. In this case the expression for the effective coupling is 47T  Kff =  7 = ^ | In (1 + C V / M * ) +  In  • + &  (3.26)  + 5fr)  B o t h terms in the denominator have cuts in the complex-p plane. T h e square root cut starting at p = 2iM corresponds to the pair production of fermions; this is familiar from the commutative model.  However the logarithmic cut in the first term, present only  when C is nonvanishing, is new to this double-scaling limit. T h i s cut could correspond to the pair creation of some nonlocal solitons present in the noncommutative  model  which survive this double-scaling limit. Unfortunately no construction of these solitons has been found.  3.5  Three Dimensions  In three dimensions the coupling has the dimensions of inverse mass, so it is not renormalizable in the small-coupling expansion.  B u t in the lavge-N expansion the theory is  renormalizable, at least near the second-order phase transition. Near this phase transition the coupling constant acquires a large anomalous dimension, rendering it effectively dimensionless and marginally renormalizable.  Chapter 3. Noncommutative  Gross-Neveu  Theory  39  T h e gap equation is now  M ( \ .A  T ^ e " ^ ) = 0 16TT  (3.27)  so that either M = 0 or M satisfies A  A  2M  .  -e~~ 16TT  (3.28)  analogous to the solutions in two dimensions. B u t while in two dimensions the M = 0 solutions is never stable, here its stability depends on the coupling. For  A < f  (3.29)  vanishing M is the stable solution, the fermions remain massless, and the chiral symmetry is unbroken. Otherwise the fermions acquire a mass according to (3.28) and the chiral symmetry is spontaneously broken. These two phases are separated by the second-order phase transition discussed above.  3.5.1  The Commutative Theory  Massless, S y m m e t r i c Phase When A <  I671-/A the  four-point function is  r  c 2 2  = -  — + H 16TT 16  (3.30)  V  1  T h e effective coupling is then given by  X  e  f  f  =  I _ JL+ H ' A  16TT  ( ' 3  3 1  )  16  If A is tuned to be close to but still less than the critical value, 1 A u —= h—, A 16TT 16'  ,„ .  2  (3.32)  v  ;  Chapter 3. Noncommutative  Gross-Neveu  40  Theory  then the theory is renormalized and the effective coupling becomes  ^ ff e  A t the critical point, when p  2  =  2 i i i• p + \p\  (3.33)  2  = 0, this becomes a nontrivial conformal field theory  whose scaling properties can be found in the 1/TV expansion. T h e condensate {^ ip ^j %  has  %  conformal dimension 1, instead of the classical value of 2, at this point. Note also from (3.32) that as the cutoff is removed along fixed p, , the bare coupling 2  flows to zero. T h i s is what happens in asymptotically free theories.  T h i s is important  because it is a major qualitative feature which the large-iV expansion of the Gross-Neveu model shares with the Yang-Mills theory which arises directly from string theory.  Massive, Broken-Symmetry Phase In this phase A > 167r/A and the correlation function is  To = 2  AM +p — ~~ + , i 16TT 4TT|P| A  e  2  2M  A  2  T1 b l  tan"  2M  , V  n  n  l  .  (3.34) '  and, using the gap equation (3.28) to remove the cutoff and bare coupling,  X E F F  =  4M2 2 . +P  ! bp  ( 3  3 5  )  So here the coupling vanishes like 16/|p| for large momenta, and it has a finite constant limit 4n/M  3.5.2  for small momenta.  Noncommutative Theory  Massless, Symmetric Phase As now expected,  there are two contributions to r | ; one is half of the  commutative  result and comes from a planar diagram, while the other is nontrivially modified by  Chapter 3. Noncommutative  Gross-Neveu  Theory  41  nonplanarity. In the massless phase the latter contribution reduces to  h  ~ The  1  32  i  +  + h)  ^ ) - (-WIWV=(^)  • (3-36)  effective coupling is now given by  1  Xeff  12  A , |<?| X  327T  16  1 327T  f\q8\  1 \ ~ \  2  \q\ /-i  da  4  \  (3.37) where the cutoff dependence has not been removed. T h e cutoff dependence in the fourth term is useful as an infrared regulator, but it is the second term which prevents the A —>• oo limit from being taken. In an attempt to renormalize this coupling, it is appropriate to introduce  £ =  (3  -  38)  since this will cancel the cutoff dependence at zero momentum. T h i s is the smallest that A can be taken as while keeping X ff e  positive since q = 0 is a m i n i m u m of the expression  above. T h i s renormalized coupling must remain positive to satisfy the phase conditions, since taking j- to zero takes the theory to the critical point. W i t h this definition  X ff e  X  r  16  327T v  1  J  A -  n  V  \ " 1  4  A  '  5  +£-  f  1  , V  '  **  w(-\q9\\q\Ja(l-aj).  s  (3.39) F r o m this expression it is clear that there is no Landau pole since A / / is positive for all e  momenta. However, the cutoff dependence has not been removed, and for large cutoff 327T  Kff  ~  If the cutoff dependence were removed at finite momentum through  (3.40)  Chapter 3. Noncommutative  Gross-Neveu  Theory  42  then the theory becomes unstable, with tachyonic modes appearing for momenta near \q9\ « A /167r. r  Hence this theory is not renormalizable, becoming trivial in the large-A limit.  T h e Double-Scaling Limit Another double-scaling limit can be defined in the three-dimensional case, if 9°  vanishes  l  and 9* = e 9 for spatial i, j . j  lj  Here e  JJ  represents the L e v i - C i v i t a tensor with e  Now take the infinite-cutoff limit while keeping A # 3  units of inverse momentum, as did C/M  2  = C  constant.  = 1.  01  Note that C  has  defined in the two-dimensional case.  So 1 X  =  eff  q C'q* + - + ——. X 8 2567T 1  (3.42)  T  v  ;  T h i s means that the particles mediating the force between the fermions has a nonrelativistic dispersion relation given by  W= I  ^V ^ ' ?+  (3  '  43)  T h i s is a long-range force, but it is not scale invariant. Even though the noncommutativity occurs at distances scales beyond that of the U V cutoff, it still influences the model because of the U V / I R mixing. T h i s was seen in the two-dimensional case as well. T h i s double-scaling limit has important implications for the uncertainty relations. For consider the original noncommutativity relation Ax Ay  = 9 =  and so Ay are finite then Heisenberg's Uncertainty Principle AxA  Px  Ap  • (C'/A ) / . 3  x  1  2  (C'/A ) / . 3  «  1  2  If Aa;  1 gives Ay  T h i s is shorter than the usual short-distance cutoff 1/A.  «  Thus the  noncommutative uncertainty relation should be invisible in the cutoff theory. Following the same analysis in the two-dimensional theory shows that in that case the commutators would be of the same order and therefore marginally observable.  Chapter 3. Noncommutative  Gross-Neveu  Theory  43  Massive, Broken-Symmetry Phase In this phase the gap equation (3.28) must be satisfied to get a stable solution. T h e cutoff dependence of the bare coupling A is also determined by the gap equation, but the cutoff dependence will not completely cancel in A / / since this theory is nonrenormalizable. e  F r o m the earlier calculations of the contributions to r | it is clear that for large momenta, above 1 / A , the dependence on momentum in  X ff is small compared to the term of order e  A. T h e effective coupling is then given approximately by  1  (3.44)  Therefore in three dimensions the effective coupling goes to zero linearly as the cutoff is taken to infinity, leaving behind the trivial theory. T h i s is even more severe than the two dimensional case, which ran to zero as an inverse logarithm of the coupling.  Chapter 4  Conclusions  Bosonic and fermionic O(N)  models with quartic interactions were studied in the large-A ' 7  limit. In the noncommutative case there are two possible cyclically inequivalent orderings of the the fields in the quartic interaction, but it is argued that the theories are consistent even when only one interaction is considered. Progress was only made for the theories containing just the factorable interaction. T h i s allowed the introduction of a nondynamical auxiliary field which facilitated a study of the phase structure and running couplings of the theories. A momentum-independent solution was assumed for the four-dimensional bosonic model. It was found to lead to a stable but nonrenormalizable theory. In the commutative case this theory is also sensible only with a fixed cutoff, but in the noncommutative case the nonrenormalizability is less severe. In two and three dimensions the fermionic Gross-Neveu model was found to be nonrenormalizable, unlike its behaviour in the commutative case. In commutative theories, even massless ones, a mass scale is generated which cuts off the coupling as it runs to large values in the infrared.  However, the noncommutativity has prevented mass from  solving the IR divergence. Since the noncommutativity blurs space, it could naively be proposed as a way to solve the U V divergences and nonrenormalizability of quantum field theory.  A more  careful analysis has shown that this need not be the case. It had already been shown  44  Chapter 4.  Conclusions  45  [15] that the U V divergences of the commutative theories would still be present in their noncommutative counterparts, but here it has been shown that renormalizability can be made even more difficult with the introduction of 6.  Bibliography  P. H o and Y . W u , "Noncommutative geometry and D-branes," Phys. Lett. B 3 9 8 , 52 (1997) [hep-th/9611233]. A . Connes, M . R. Douglas and A . Schwarz, "Noncommutative geometry and matrix theory: Compactification on tori," J H E P 9802, 003 (1998) [hep-th/9711162]. C . C h u and P. Ho, "Constrained quantization of open string in background B field and noncommutative D-brane," Nucl. Phys. B 5 6 8 , 447 (2000) [hep-th/9906192]. N . Seiberg and E . Witten, "String theory and noncommutative geometry,"  JHEP  9909, 032 (1999) [hep-th/9908142]. T . Lee, "Canonical quantization of open string and noncommutative  geometry,"  Phys. Rev. D 6 2 , 024022 (2000) [hep-th/9911140]. M . Laidlaw, "Noncommutative geometry from string theory: Annulus corrections," hep-th/0009068. J . Gomis and T . Mehen, "Space-time noncommutative field theories and unitarity," hep-th/0005129. J . Zinn-Justin, C h 1.2, "Quantum Field Theory and Critical Phenomena," 2  n d  ed.  Oxford: Clarendon Press, 1993. See C h . 6, [8] See C h . 28, [8] S. Coleman, "1/N," Presented at 1979 International School of Subnuclear Physics, "Pointlike Structures Inside and Outside Hadrons." A . Connes, "Noncommutative Geometry," Academic Press, 1994. R. Szabo, "Quantum field theory on noncommutative spaces," Lecture given at A P C T P - K I A S Winter School, "Strings and D-Branes 2000." M . Douglas and N . Nekrasov, "Noncommutative Field Theory," hep-th/0106048 T . Filk, "Divergences in a field theory on quantum space," Phys. Lett. B 3 7 6 , 53 (1996). I. Chepelev and R. Roiban, "Renormalization of quantum field theories on noncommutative R * * d . I: Scalars," J H E P 0005, 037 (2000) [hep-th/9911098]. I. Chepelev and R. Roiban, "Convergence theorem for non-commutative Feynman graphs and renormalization," hep-th/0008090.  46  Bibliography  47  [18] S. Minwalla, M . V a n Raamsdonk and N . Seiberg, "Noncommutative perturbative dynamics," hep-th/9912072. [19] M . V a n Raamsdonk and N . Seiberg, "Comments on noncommutative perturbative dynamics," J H E P 0003, 035 (2000) [hep-th/0002186]. [20] W . A . Bardeen and M . Moshe, "Phase structure of the O(N)  vector model," Phys.  Rev. D 2 8 , 06 1372 (1983). [21] S. S. Gubser and S. L . Sondhi, "Phase structure of non-commutative scalar field theories," hep-th/0006119. [22] E . T . Akhmedov, P. DeBoer, G . W . Semenoff,  "Running couplings and triviality  of field theories on non-commutative spaces," T o appear in Phys. Rev. D [hepth/0010003]. [23] E . T . Akhmedov, P. DeBoer, G . W . Semenoff,  "Non-commutative Gross-Neveu  model at large N , " J H E P 0106 (2001) 009 [hep-th/0103199].  


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