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Path integrals for multiply connected spaces Scholte-van de Vorst, Matthew 2009

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Path Integrals for Multiply Connected Spaces by Matthew Scholte-van de Vorst  B.Sc., University of Manitoba, 2006  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in The Faculty of Graduate Studies (Physics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) August 2009 c Matthew Scholte-van de Vorst 2009  Abstract We derive the propagator for a particle constrained to a torus and to a Klein Bottle. This is accomplished by considering relative symmetries between the desired system and a system for which the propagator is known. This result is checked against the propagator derived via the method of stationary state construction, for which the entire spectrum of the Hamiltonian is required. We also briefly consider the application of further constraints to the systems, and the implications of different symmetries on the same constraint.  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Tables  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  vi  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2 Circle Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Circle Stationary State Propagator . . . . . . . . . . . . . . . 2.2 Circle Symmetry Propagator . . . . . . . . . . . . . . . . . .  3 3 5  3 Torus and Klein Bottle Foundations . . . . . . . . . . . . . . 3.1 Torus Foundations . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Klein Bottle Foundations . . . . . . . . . . . . . . . . . . . .  7 7 8  4 Torus Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Torus Symmetry Propagator . . . . . . . . . . . . . . . . . . 4.2 Torus Stationary State Propagator . . . . . . . . . . . . . . .  12 12 13  5 Klein Bottle Propagator . . . . . . . . . . . . . . . . . . . . . . 5.1 Klein Bottle Symmetry Propagator . . . . . . . . . . . . . . . 5.2 Klein Bottle Stationary State Propagator . . . . . . . . . . .  15 15 18  6 Additional Constraints . . . . . . . . . . . . . . . . . . . . . . . 6.1 Constraint Applied to Torus . . . . . . . . . . . . . . . . . . . 6.2 Constraint Applied to Klein Bottle . . . . . . . . . . . . . . .  23 23 24  iii  Table of Contents 7 Conclusion  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  27  A Jacobi Theta Function Identities . . . . . . . . . . . . . . . .  28  iv  List of Tables 3.1  Klein Bottle eigenstates and eigenvalues. . .  . . . . . . . . . .  9  v  List of Figures 3.1 3.2  Torus Construction . . . . . . . . . . . . . . . . . . . . . . . . Klein Bottle Construction . . . . . . . . . . . . . . . . . . . .  7 9  vi  Acknowledgements My thanks to the faculty and staff of UBC who have assisted me throughout my degree, especially Dr. van Raamsdonk. I thank my supervisors Drs. Schleich and Witt whose questions, answers, and feedback have been invaluable in the completion of both my thesis and degree as a whole. Special thanks to my family for their support and encouragement over the years.  vii  Dedication To Oma & Opa  viii  Chapter 1  Introduction The propagator is a powerful tool in quantum mechanics, as knowing the propagator for a system means knowing the solution to the time dependent Schrodinger equation. If we know the state ψ(x0 , t0 ) at some initial point (x0 , t0 ), then the state at some later point (x, t) is given by: ψ(x, t) = G(x, t; x0 , t0 )ψ(x0 , t0 )  (1.1)  where G(x, t; x0 , t0 ) is the propagator associated with the system. Path integrals arise in determining the form of the propagator: Schulman [6] derives the propagator for a free particle with time independent Hamiltonian (which allows us to set t0 = 0) to be:  lim  G(x, t; x0 , 0) =N →∞ (  where xN +1 = x and form:  m (N +1)/2 ) 2πi  dx1 . . . dxN exp[  im 2  N  (xj+1 − xj )2 ] j=0  (1.2) = t/(N + 1). This can be simplified to the closed  G(x, t; x0 , 0) =  m im exp[ (x − x0 )2 ] 2πi t 2 t  (1.3)  where the phase is the classical action divided by . Determining the propagator becomes more complicated on multiply connected spaces, as it is possible to generate two paths having the same endpoints which cannot be continuously deformed into each other. In this situation, it is useful to define a winding number which separates these paths into homotopy classes, within which it is possible to perform this deformation continuously. However, care must be taken in determining the propagator to account for these winding numbers. These complications mean that one must often return to the definition in order to account for them properly. One method of determining the propagator in any space is to perform a sum over states. If (ψn (x), En ) is a pair of stationary states and ener1  Chapter 1. Introduction gies satisfying the Hamiltonian for the space, then the propagator can be expressed as: ∞  ψn (xf )ψn∗ (xi )exp(−iEn t)  G(xf , t; xi , 0) =  (1.4)  n=−∞  However, it is possible that simplifying the propagator to a useful form from this expression can still be quite complicated, as performing the sum on multiply connected spaces is not straight-forward. One might wish to obtain a simpler procedure to construct the propagator more quickly, without having to determine explicitly the entire spectrum for the Hamiltonian. Alternatively, one can make use of the homotopy classes as outlined in various papers [2, 4, 5]. If one can find a projection p to the desired space ˜ , the problem becomes much M from a simply connected covering space M ˜ from simpler. Paths in M from m to n (m, n ∈ M ) correspond to paths in M −1 −1 some fixed m ˜ ∈ p (m) to each n ˜ i ∈ p (n), where i runs over all pre-images ˜ (using the projection p to of n. Then we carry out the path integral in M carry over the Lagrangian from M ) and sum over i, including phase factors determined by the system. This is one method we will use to determine the propagator for the circle. If we can relate the desired space to a simpler one (not necessarily a simply connected covering space) through various symmetries that are either preserved or lost between the two spaces, perhaps it is possible to construct the propagator without solving for the entire spectrum or concerning ourselves with mapping to a covering space. In this fashion, we will construct the propagator for a particle constrained to a torus and to a Klein Bottle, beginning with the propagator for a particle constrained to a circle. As a check, we will also explicitly construct the propagator from the spectrum using the method of stationary states (1.4). We will also examine briefly the question of applying further constraints to these systems which correspond to the particle model of a gauge theory.  2  Chapter 2  Circle Propagator 2.1  Circle Stationary State Propagator  We begin by considering the derivation of the propagator for a particle on the circle S 1 presented by Schulman [5]. This will facilitate the symmetry construction of the torus propagator, as the torus is the tensor product of two circles. Construction of the circle is achieved by identification of x = 0 and x = 2π on the interval [0, 2π]. Alternatively, the circle is constructed via the identification x = x + 2π on the real line , which is more convenient as it preserves the geometry of . On the circle, the local Schrodinger equation for a free particle is (taking =1, M=1): 1 − ∂x2 ψ(x) = Eψ(x) (2.1) 2 The condition that the system is on the circle is enforced by requiring the state ψ(x) also obey the boundary condition: 1  S S ψ(x) = ψ(x + 2π)  (2.2)  S1  where S is a unitary symmetry operator which commutes with the Hamil1 tonian, [S S , H] = 0. Hence, ψ(x) can be simultaneously diagonalized with respect to both operators. If we write Sψ = cψ (where superscripts and arguments have been suppressed), then since: S n ψ(x) = ψ(x + 2nπ)  (2.3)  S n S m = S n+m  (2.4)  and:  it follows that c = eiσ (for σ a real constant) and Bloch’s theorem states that:  3  Chapter 2. Circle Propagator  iσx )u(x) (2.5) 2π where u(x) = u(x + 2π) is a periodic function. The solution of Schrodinger’s equation (2.1) for ψ(x) of this form yields the complete set of eigenstates: ψ(x) = exp(  1 iσx 1 ψnS (x) = √ exp(inx + ) 2π 2π  (2.6)  σ 2 1 1 ) (2.7) EnS = (n + 2 2π Thus, instead of a unique quantization, one obtains a set of inequivalent quantizations parameterized by σ, and perfect periodicity corresponds to the unique quantization σ = 0. As a result, stationary state construction of the propagator (1.4) takes the form: 1  GS (xf , t; xi , 0) = n  1 iσx it σ 2 exp(inx + − (n + ) ) 2π 2π 2 2π  (2.8)  where x = xf − xi , and the propagator argument shall henceforth be suppressed. In order to facilitate comparison with the propagator as derived from symmetry considerations, it is useful to introduce the Jacobi theta function defined by: ∞  exp(iπn2 T + 2inZ)  Θ3 (Z, T ) =  (2.9)  n=−∞  As a trivial result of the definition, Θ3 (−Z, T ) = Θ3 (Z, T ). Furthermore, two useful identities of the theta function are: Θ3 (Z + aπ + bπT, T ) = exp(−iπb2 T − 2ibZ)Θ3 (Z, T ) Θ3 (Z, T ) = (−iT )−1/2 exp(Z 2 /iπT )Θ3 (Z/T, −1/T )  (2.10)  (2.11)  for (a, b) ∈ Z, the proof of which can be found in Appendix A. Substituting the theta function (2.9) yields: 1  GS =  1 σx σ2t x σt t exp(i( − 2 ))Θ3 ( − ,− ) 2π 2π 8π 2 4π 2π  (2.12)  4  Chapter 2. Circle Propagator which, if we apply the Jacobi identity (2.11) becomes:  1  1 t σx σ2t (i )−1/2 exp(i( − 2 ))∗ 2π 2π 2π 8π σt 2 t x σt t 2π x ) /iπ(− ))Θ3 (( − )/(− ), ) exp(( − 2 4π 2π 2 4π 2π t 1 1/2 ix2 σ πx 2π =( ) exp( )Θ3 ( − , ) 2πit 2t 2 t t  GS =  2.2  (2.13)  Circle Symmetry Propagator  As a comparison with the stationary state construction, we can also use symmetry considerations to construct the propagator. Since two paths having the same winding number and fixed endpoints can be continuously deformed into each other, we write the full propagator as a sum over the winding number n: ∞ 1  1  GS (xf , t; xi , 0) =  An GSn (xf , t; xi , 0)  (2.14)  n=−∞  for some non-trivial set of An ’s. By varying one endpoint (say xf ) through a full circle, we find that the full propagator is unchanged up to a phase 1 1 factor, while we have translated GSn → GSn−1 . This, combined with (2.14), tells us that: An+1 = eiσ An  (2.15)  Furthermore, we know that A0 = 1, and so fixing the arbitrary phase of A0 to 0 yields: 1  GS =  einσ GSn  1  (2.16)  n  where the argument of the propagator shall again be suppressed. Since one can (locally) define a smooth mapping f : → S 1 and inverse between the real line and circle for fixed endpoints, for each value of the winding number the propagator is that of a free particle (1.3). Substituting 1 for GSn into (2.16) yields: 1  GS =  ( n  1 1/2 i(x − 2πn)2 ) exp(inσ + ) 2πit 2t  (2.17)  5  Chapter 2. Circle Propagator Substituting the theta function (2.9) into our expression for the propagator on the circle (2.17) yields: ix2 σ πx 2π 1 1/2 ) exp( )Θ3 ( − , ) 2πit 2t 2 t t in agreement with the stationary state propagator (2.13). 1  GS = (  (2.18)  6  Chapter 3  Torus and Klein Bottle Foundations 3.1  Torus Foundations  Figure 3.1: Torus Construction We begin by generating the torus T 2 from a rectangle of side lengths a and b, seen in Fig. 3.1. Opposite sides (of equal length) are identified and connected, with the x-direction denoting motion parallel to side length b and y-direction parallel to side length a. In this way we construct a torus with boundary conditions:  (x, y) = (x, y + a) (x, y) = (x + b, y)  (3.1)  Locally, Schrodinger’s equation for a free particle is just: 7  Chapter 3. Torus and Klein Bottle Foundations  1 − (∂x2 + ∂y2 )ψ = Eψ (3.2) 2 Eigenstates (non-normalized) and corresponding eigenvalues obeying the boundary conditions (3.1) are: 2  T ψrs = exp(2πi(rx/b + sy/a))  (3.3)  2πs 2 1 2πr 2 T2 ) +( ) ) Ers = (( 2 b a  (3.4)  2  2  T (x, y) = ψ T (x + nb, y + ma) for (n, m) ∈ Z. for (r, s) ∈ Z, where ψrs rs However, as seen on the circle, perfect periodicity corresponds to the specific quantization in which σ = 0. The more general quantization is implemented by unitary symmetry operators generated by the fundamental region:  2  SxT f (x, y) = f (x + b, y) 2  SyT f (x, y) = f (x, y + a)  (3.5) 2  Simultaneous normalized eigenstates and eigenvalues of H, SxT , and SyT are then: iσx iδy 1 T2 + ) ψrs = √ exp(2πi(rx/b + sy/a) + b a ab  2  (3.6)  1 2πr + σ 2 2πs + δ 2 T2 Ers = (( ) +( ) ) (3.7) 2 b a for (r, s) ∈ Z, where σ and δ are the phases picked up in completing a full circuit in the x and y directions respectively (eg: x goes to x+b).  3.2  Klein Bottle Foundations  Beginning with the same identifications on the rectangle as for the torus, we join opposite ends. However, to obtain the Klein Bottle K 2 we introduce a twist before joining the ends, as shown in Fig. 3.2. Note that the construction used here is non-standard: the second twist has been included to introduce symmetry between x and y, whereas the standard (one twist) construction is asymmetrical (the untwisted direction behaves similarly to 8  Chapter 3. Torus and Klein Bottle Foundations  Figure 3.2: Klein Bottle Construction the torus). For this construction, the boundary conditions are:  (x, y) = (−x, y + a) (x, y) = (x + b, −y)  (3.8)  Locally, Schrodinger’s equation is the same for the Klein Bottle as for the torus (3.2). However, solutions are much different as a result of the boundary conditions. In terms of restricted integers (k,l) there are four groups of solutions given in Table 3.1. K ψkl  2  πly cos( πkx b )cos( a ) πly sin( πkx b )cos( a ) πly cos( πkx b )sin( a ) πly sin( πkx b )sin( a )  K Ekl 2  π k 2 2 (( b ) π2 k 2 2 (( b ) π2 k 2 2 (( b ) π2 k 2 2 (( b )  2  + ( al )2 ) + ( al )2 ) + ( al )2 ) + ( al )2 )  k parity l parity even even odd odd  even odd even odd  Table 3.1: Klein Bottle eigenstates and eigenvalues. Comparison of the Klein Bottle and torus states is easiest if we rewrite the torus states in terms of the restricted integers (k,l):  9  Chapter 3. Torus and Klein Bottle Foundations  2  T ψkl = exp(πi(kx/b + ly/a))  (3.9)  π2 k 2 l (( ) + ( )2 ) (3.10) 2 b a where k and l are both even. Comparing this to Table 3.1, we see that the energy spectrum has the same form for both spaces with a denser spectrum for the Klein Bottle. Expanding the exponential states for the torus, we see that the four states of the Klein Bottle are all present, but with different energies. From this, we can see that the particle on the Klein Bottle is the same as on the torus, but with the degeneracy in energy lifted by the symmetric twists introduced in the construction. Before continuing, we first simplify the Klein Bottle states into a compact form, allowing us to lift the restrictions on k and l. In order to do so, we first note that 1 + (−1)k evaluates to 0 if k is odd, and 2 if k is even. Applying this, the states can be written as: 2  T Ekl =  2  K ψkl = sin(  πkx π πly π + (1 + (−1)l ))sin( + (1 + (−1)k )) b 4 a 4  (3.11)  while the energies remain unchanged. As with the circle and torus before, this is the specific quantization assuming perfect periodicity. The unitary symmetry operators generated by the fundamental region are: 2  SxK f (x, y) = f (x + b, −y) 2  SyK f (x, y) = f (−x, y + a)  (3.12) 2  Simultaneous normalized eigenstates and eigenvalues of H, SxK , and Sy are then: K2  2  K ψkl =  4 πkx σx π sin( + + (1 + (−1)l ))∗ ab b 2b 4 πly δy π sin( + + (1 + (−1)k )) a 2a 4  1 πk σ πl δ K2 Ekl = (( + )2 + ( + )2 ) 2 b 2b a 2a  (3.13)  (3.14) 10  Chapter 3. Torus and Klein Bottle Foundations for (k, l) ∈ Z, where σ and δ are the phases picked up in completing two full circuits in the x and y directions respectively (eg: x goes to x+2b).  11  Chapter 4  Torus Propagator 4.1  Torus Symmetry Propagator  Consider a particle on the torus, which is subject to no further constraints. We wish to find the propagator describing that particle’s motion: as with the circle it is possible to construct this propagator by a purely symmetrybased argument. Namely, consider the particle to be further constrained to travel in only the x-direction. The propagator, under this further constraint, should be identical to that of the particle on a circle, for this new constraint physically replicates exactly that. Likewise, constraining the particle to travel in the y-direction should also yield the propagator for a circle. This is reflected in the decomposition of the torus space, which can be expressed as T 2 = S 1 × S 1 . Generalizing the derivation of the circle propagator with this in mind, we can express the torus propagator as a sum over winding numbers in each direction: 2  2  GT =  Ar Bs GTrs  (4.1)  r,s  where the coefficients, as for the circle, become phase factors and the individual propagators those of a free particle. Since we are now dealing with the general period b, we must rescale the coordinate by 2π b in the free particle state, carrying it through to the final form of the propagator. The modified states (2.6), energies (2.7), and finally propagator (2.18) become:  1  1 iσx 2π 1 ψnS = √ exp(i nx + ) b b b  (4.2)  1 2πn σ 2 1 EnS = ( + ) 2 b b  (4.3)  GS = (  1 1/2 ix2 σ xb b2 ) exp( )Θ3 ( − , ) 2πit 2t 2 2t 2πt  (4.4)  12  Chapter 4. Torus Propagator Making these substitutions into the torus propagator (4.1):  2  GT = r,s  1 i xb ya ib2 r2 ia2 s2 exp( (x2 + y 2 ) + ir(σ − ) + is(δ − ) + + ) 2πit 2t t t 2t 2t (4.5)  which, when we substitute the theta function (2.9), becomes:  2  GT =  i σ xb b2 δ ya a2 1 exp( (x2 + y 2 ))Θ3 ( − , )Θ3 ( − , ) 2πit 2t 2 2t 2πt 2 2t 2πt  (4.6)  We recognize this as simply being the product of two propagators for the circle, as we expect given the relationship between the two spaces.  4.2  Torus Stationary State Propagator  Now, as with the circle, we can check our result for the symmetry propagator (4.6) against the stationary state form. Beginning from the definition for the propagator as a sum over stationary states: ∞ ∗ ψrs (xf , yf )ψrs (xi , yi )exp(−iErs t)  G(xf , yf , t; xi , yi , 0) =  (4.7)  r,s=−∞  and substituting the normalized states (3.6) and energies (3.7), we find that the propagator for the torus is: 2  GT = r,s  iσx iδy 1 exp(2πi(rx/b + sy/a) + + ab b a −  it 2πr + σ 2 2πs + δ 2 (( ) +( ) )) 2 b a  (4.8)  where x = xf − xi , y = yf − yi , and the argument of the propagator shall once more be suppressed. Substituting the theta function (2.9) into the above propagator (4.8) yields:  13  Chapter 4. Torus Propagator  2  1 iσx iδy itσ 2 itδ 2 exp( + − 2 − 2 )∗ ab b a 2b 2a πx πσt 2πt πy πδt 2πt Θ3 ( − 2 , − 2 )Θ3 ( − 2 ,− 2 ) b b b a a a  GT =  (4.9)  Applying the Jacobi identity (2.11), we find: 1 2πit −1/2 2πit −1/2 iσx iδy itσ 2 itδ 2 ( 2 ) ( 2 ) exp( + − 2 − 2 )∗ ab b a b a 2b 2a πx πσt 2 2π 2 it πy πδt 2 2π 2 it exp(( − 2 ) / − 2 )exp(( − 2 ) /− )∗ b b b a a a2 2πt b2 πy πδt 2πt a2 πx πσt − 2 )/ − 2 , )Θ3 (( − 2 )/ − 2 , ) Θ3 (( b b b 2πt a a a 2πt  2  GT =  (4.10)  which simplifies to:  2  GT =  1 i σ xb b2 δ ya a2 exp( (x2 + y 2 ))Θ3 ( − , )Θ3 ( − , ) 2πit 2t 2 2t 2πt 2 2t 2πt  (4.11)  in agreement with the symmetry propagator (4.6).  14  Chapter 5  Klein Bottle Propagator 5.1  Klein Bottle Symmetry Propagator  Now that we have a procedure for deriving the propagator through symmetry considerations, let us apply it to the Klein Bottle and compare with the stationary state result. We begin by recalling the symmetry for the Klein Bottle (3.12), and notice that two successive applications of the same symmetry operator looks like a single application of the torus symmetry (3.5) with twice the period: 2  2  (SxK )2 f (x, y) = SxK f (x + b, −y) = f (x + 2b, y)  (5.1)  Since the Klein Bottle chosen is symmetric in x and y, it suffices to discuss only one direction: the other follows immediately. This suggests that the Klein Bottle propagator should contain a similar structure as the torus propagator with twice the period, which we will call G´T 2 . If we take 2 as an initial guess some combination of G´T 2 and S K , we can check certain properties. First, the propagator should only pick up a phase when the symmetry operator is applied (namely σ/2, since that would produce the required phase for two applications, ie: x goes to x+2b): 2  2  2  SxKf GK (xf , yf , t; xi , yi , 0) = exp(iσ/2)GK (xf , yf , t; xi , yi , 0)  (5.2)  and second, the propagator is unitary (ie: the propagator should preserve the norm of the states), since we can write: 2  2  2  K K ψkl (xf , yf , t) = GK (xf , yf , t; xi , yi , 0)ψkl (xi , yi , 0)  (5.3) 2  In satisfying the first property (5.2), let us successively apply S K to 2 ´ GT until we are left with a phase factor:  15  Chapter 5. Klein Bottle Propagator  2 2 SxKf G´T 2 =SxKf  r,s  = r,s  = r,s  1 i exp( ((x − 2rb)2 + (y − 2sa)2 ) + irσ + isδ) 2πit 2t  1 i exp( ((x + b − 2rb)2 + (−¯ y − 2sa)2 ) + irσ + isδ) 2πit 2t 1 i exp( ((x − b − 2(r − 1)b)2 + (−¯ y − 2sa)2 )+ 2πit 2t  iσ + i(r − 1)σ + isδ) 1 i =exp(iσ) exp( ((x − b − 2rb)2 + (−¯ y − 2sa)2 )+ 2πit 2t r,s irσ + isδ)  2 2 (SxKf )2 G´T 2 =(SxKf )2  r,s  = r,s  = r,s  (5.4)  i 1 exp( ((x − 2rb)2 + (y − 2sa)2 ) + irσ + isδ) 2πit 2t  i 1 exp( ((x + 2b − 2rb)2 + (y − 2sa)2 ) + irσ + isδ) 2πit 2t 1 i exp( ((x − 2(r − 1)b)2 + (y − 2sa)2 )+ 2πit 2t  iσ + i(r − 1)σ + isδ) 1 i =exp(iσ) exp( ((x − 2rb)2 + (y − 2sa)2 ) + irσ + isδ) 2πit 2t r,s =exp(iσ)G´T 2  (5.5)  where y¯ = yf + yi . Therefore, as an initial propagator, take: 2 2 2 GK =4(G´T 2 + exp(−iσ/2)SxKf G´T 2 + exp(−iδ/2)SyKf G´T 2 + 2 2 exp(−iσ/2)exp(−iδ/2)SxKf SyKf G´T 2 )  (5.6)  where the norm has been chosen to satisfy the unitarity property (5.3), and the symmetry property (5.2) is obeyed as a result of (5.5):  16  Chapter 5. Klein Bottle Propagator  2 2 2 2 2 2 SxKf GK =4(SxKf G´T 2 + exp(−iσ/2)(SxKf )2 G´T 2 + exp(−iδ/2)SxKf SyKf G´T 2 + 2 2 exp(−iσ/2)exp(−iδ/2)(SxKf )2 SyKf G´T 2 ) 2 2 2 =4(SxKf G´T 2 + exp(iσ/2)G´T 2 + exp(−iδ/2)SxKf SyKf G´T 2 + 2 exp(iσ/2)exp(−iδ/2)SyKf G´T 2 )  =exp(iσ/2)GK  2  (5.7)  Substituting for the torus propagator (4.6) and applying the symmetry operators, the Klein Bottle propagator becomes:  2  GK =  i σ xb b2 δ ya a2 2 (exp( (x2 + y 2 ))Θ3 ( − , )Θ3 ( − , )+ πit 2t 2 2t 2πt 2 2t 2πt iσ i exp( ((x + b)2 + (−¯ y )2 ) − )∗ 2t 2 σ (x + b)b b2 δ y¯a a2 Θ3 ( − , )Θ3 ( + , )+ 2 2t 2πt 2 2t 2πt i iδ exp( ((−¯ x)2 + (y + a)2 ) − )∗ 2t 2 σ x ¯b b2 δ (y + a)a a2 Θ3 ( + , )Θ3 ( − , )+ 2 2t 2πt 2 2t 2πt i iσ iδ exp( ((−¯ x + b)2 + (−¯ y + a)2 ) − − )∗ 2t 2 2 σ (¯ x − b)b b2 δ (¯ y − a)a a2 Θ3 ( + , )Θ3 ( + , )) (5.8) 2 2t 2πt 2 2t 2πt  Applying the boundary conditions (3.8) associated with the symmetry operators to the final endpoints, we can transform x + b = (xf + b) − xi −¯ y = −yf − yi  →  xf − xi = x yf − yi = y  (5.9)  Applying this transformation to individual terms in the propagator (5.8) and factoring yields:  17  Chapter 5. Klein Bottle Propagator  2 −iσ −iδ iσ iδ (1 + exp( ) + exp( ) + exp(− − ))∗ πit 2 2 2 2 2 i 2 σ xb b δ ya a2 exp( (x + y 2 ))Θ3 ( − , )Θ3 ( − , ) 2t 2 2t 2πt 2 2t 2πt  2  GK =  5.2  (5.10)  Klein Bottle Stationary State Propagator  Now, as with the torus, we substitute the normalized states (3.13) and energies (3.14) into the stationary state expression for the propagator (4.7) which yields: 2  GK = k,l  sin(  πkxf σxf 4 π sin( + + (1 + (−1)l ))∗ ab b 2b 4 πkxi σxi π K2 + + (1 + (−1)l )) ∗ ((Y )) ∗ exp(−iEkl t) b 2b 4  (5.11)  where ((Y)) denotes the equivalent function of x for y (y replaces x, l replaces k, δ replaces σ, etc.). Applying the product formula for sine, we have: 2  GK = k,l  1 πk¯ x σ¯ x π (cos( + + (1 + (−1)l ))− ab b 2b 2 cos(  πkx σx K2 + )) ∗ ((Y )) ∗ exp(−iEkl t) b 2b  (5.12)  Rewriting as exponentials via Euler’s formula, and using: π exp(i (1 + (−1)l )) = exp(iπ(l + 1)) 2 the propagator becomes:  (5.13)  18  Chapter 5. Klein Bottle Propagator  2  GK = k,l  σ x ¯ 1 ((exp(i(πk + ) + iπ(l + 1))+ 4ab 2 b σ x ¯ exp(−i(πk + ) + iπ(l + 1)))− 2 b σ x σ x (exp(i(πk + ) ) + exp(−i(πk + ) )))∗ 2 b 2 b K2 ((Y )) ∗ exp(−iEkl t)  (5.14)  Applying exp(iπ) = −1 and cancelling the resulting overall -1 factor on both ((X)) and ((Y)), with (3.14) substituted the propagator becomes: 2  GK = k,l  σ x ¯ σ x ¯ 1 (exp(i(πk + ) + iπl) + exp(−i(πk + ) + iπl)+ 4ab 2 b 2 b σ x σ x ) ) + exp(−i(πk + ) ))∗ 2 b 2 b σ2 σπk π 2 k 2 )) ∗ ((Y )) exp(−it( 2 + 2 + 8b 2b 2b2  exp(i(πk +  (5.15)  Carrying out the cross-multiplication and substituting the theta function (2.9) yields 16 terms of similar structure, so it is useful to define the following function: σu σ 2 t δv δ2t − 2+ − 2 ))∗ 2b 8b 2a 8a −σπt πw πt −δπt πz πt Θ3 ( + , − 2 ) ∗ Θ3 ( + ,− ) 4b2 2b 2b 4a2 2a 2a2  F (u, v, w, z) =exp(i(  (5.16)  Substituting this into the propagator yields:  19  Chapter 5. Klein Bottle Propagator  2  GK =  1 (F (¯ x, y¯, x ¯ + b, y¯ + a) + F (¯ x, −¯ y, x ¯ + b, −¯ y + a)+ 4ab F (−¯ x, y¯, −¯ x + b, y¯ + a) + F (−¯ x, −¯ y , −¯ x + b, −¯ y + a)+ F (¯ x, y, x ¯, y + a) + F (¯ x, −y, x ¯, −y + a)+ F (−¯ x, y, −¯ x, y + a) + F (−¯ x, −y, −¯ x, −y + a)+ F (x, y¯, x + b, y¯) + F (x, −¯ y , x + b, −¯ y )+ F (−x, y¯, −x + b, y¯) + F (−x, −¯ y , −x + b, −¯ y )+ F (x, y, x, y) + F (x, −y, x, −y)+ F (−x, y, −x, y) + F (−x, −y, −x, −y))  (5.17)  Now, as with the symmetry propagator, we can apply the boundary conditions to transform individual terms. Recalling (5.9): x + b = (xf + b) − xi −¯ y = −yf − yi  →  xf − xi = x yf − yi = y  →  xf − xi = x yf − yi = y  We can also write: x + b = xf − (xi − b) y¯ = yf + yi  (5.18)  Applying these transformations to the propagator (5.17) yields: 2  GK =  1 (4F (x − b, y − a, x, y) + 2F (x, y − a, x, y)+ 4ab 2F (x, −y − a, x, −y) + 2F (x − b, y, x, y)+ 2F (−x − b, y, −x, y) + F (x, y, x, y)+ F (x, −y, x, −y) + F (−x, y, −x, y) + F (−x, −y, −x, −y)) (5.19)  There is one last type of transformation we can generate from applying the boundary condition twice, once on each endpoint: x = (xf + b) − (xi + b) −y = −yf + yi  →  xf − xi = x yf − yi = y  (5.20)  which allows us to further collapse the propagator into:  20  Chapter 5. Klein Bottle Propagator  2  GK =  1 (F (x − b, y − a, x, y) + F (x, y − a, x, y)+ ab F (x − b, y, x, y) + F (x, y, x, y))  (5.21)  Applying the Jacobi identity (2.11) to our function F, we can rewrite it as: σu σ 2 t δv δ2t 2ab )exp(i( − 2+ − 2 ))∗ iπt 2b 8b 2a 8a −σπt πw 2 π 2 t −δπt πz 2 π 2 t exp(i( + ) / )exp(i( + ) / 2 )∗ 4b2 2b 2b2 4a2 2a 2a −σπt πw πt 2b2 Θ3 (( + )/ − 2 , )∗ 4b2 2b 2b πt −δπt πz πt 2a2 Θ3 (( + )/ − , ) (5.22) 4a2 2a 2a2 πt  F (u, v, w, z) =(  which simplifies to: 2ab σ(u − w) w2 δ(v − z) z 2 )exp(i( + + + ))∗ iπt 2b 2t 2a 2t σ wb 2b2 δ za 2a2 Θ3 (( − ), )Θ3 (( + ), ) (5.23) 2 t πt 2 t πt  F (u, v, w, z) =(  Substituting this into our expression for the propagator (5.21) yields:  2  GK =  −σ x2 −δ y 2 2 (exp(i( + + + ))+ iπt 2 2t 2 2t x2 −δ y 2 −σ x2 y 2 exp(i( + + )) + exp(i( + + ))+ 2t 2 2t 2 2t 2t x2 y 2 σ xb 2b2 δ ya 2a2 exp(i( + )))Θ3 (( − ), )Θ3 (( + ), ) (5.24) 2t 2t 2 t πt 2 t πt  which can be rewritten as:  21  Chapter 5. Klein Bottle Propagator  2  2 −iσ −iδ iσ iδ (1 + exp( ) + exp( ) + exp(− − ))∗ πit 2 2 2 2 2 i 2 σ xb b δ ya a2 exp( (x + y 2 ))Θ3 ( − , )Θ3 ( − , ) 2t 2 2t 2πt 2 2t 2πt  GK =  (5.25)  in agreement with the symmetry propagator (5.10).  22  Chapter 6  Additional Constraints 6.1  Constraint Applied to Torus  Suppose we wish to apply additional constraints to the systems, such as the constraint: py ψ = 0  (6.1)  Imposition of such a constraint corresponds to the particle model of a gauge theory that is a simple analog of the one in [3]. Because of the phase that appears in the states we cannot simply use the old momentum operator py = −i∂y , but must instead determine a new momentum operator which includes the phase. As a first step in determining the new form, consider the effect of our old momentum operator on the state: 2πs δ T 2 + )ψrs a a Therefore a momentum operator of the form: 2  T py ψrs =(  (6.2)  1 δ = ∂y − (6.3) i a will eliminate the constant factor and allow us to satisfy the constraint (6.1) which becomes: pTy  2C  2πs T 2 ψ =0 (6.4) a rs with the condition s=0. With the new momentum operator, the energy spectrum becomes: pTy  2C  2  T ψrs =  1 2πr 2 2πs 2 T 2C Ers = (( ) +( ) ) 2 b a and the constrained states and energies become:  (6.5)  23  Chapter 6. Additional Constraints  1 2πirx iσx iδy T 2C ψr0 = √ exp( + + ) b b a ab  (6.6)  1 2πr 2 T 2C Er0 = ( ) (6.7) 2 b Note that the constrained system matches that of the circle with periodicity b, as is reasonable to expect given the nature of the constraint we’ve imposed. While the y coordinate still appears in the constrained state, the constraint implies that this term merely contributes an overall identical phase factor to the individual states. By redefining our coordinate system to shift the x-axis to position y, this phase factor can be eliminated.  6.2  Constraint Applied to Klein Bottle  Now consider applying the same constraint (6.1) to the Klein Bottle. The old momentum operator py = −i∂y acting on our eigenstates (3.13) yields:  2  4 πl δ πkx σx π ( + )sin( + + (1 + (−1)l ))∗ ab a 2a b 2b 4 πly δy π + + (1 + (−1)k )) cos( a 2a 4 πl δ K2 = − i( + )ψ˜kl a 2a  K py ψkl =−i  (6.8)  so we see that our eigenstates of the Hamiltonian are not eigenstates of the old momentum operator. 2 To see why that is, consider the commutator of py and SxK : 2  2  2  [py , SxK ]f (x, y) =(−i∂y SxK + SxK i∂y )f (x, y) 2 ∂f (x, y) = − i∂y f (x + b, −y) + iSxK ∂y ∂f (x + b, −y) ∂f (x + b, −y) =−i −i =0 ∂y ∂y  (6.9)  Since the operators do not commute, unlike on the torus, it is not possible to simultaneously diagonalize the states. Eigenstates of the Hamiltonian must also be eigenstates of the symmetry operator, so it is not possible to form eigenstates of the momentum operator. As such, we cannot impose 24  Chapter 6. Additional Constraints the constraint (6.1) on the Klein Bottle, since any candidate for a new momentum operator would also not commute with the symmetry operator.  25  Chapter 7  Conclusion By considering relative symmetries between two systems, one for which the propagator is known and one for which we wish to find the propagator, we have derived the desired propagator without applying any knowledge of the states of the target system. For the torus, we began with the circle and the knowledge that in both the x and y directions, motion around the torus replicates motion around the circle. For the non-standard (double-twist) Klein Bottle, we began with the torus and the knowledge that motion twice around the Klein Bottle in either direction replicates motion once around the torus. Armed with these relationships and the symmetries obeyed on each space, we have found the propagator for the torus and Klein Bottle. In each case, we have checked our result against the stationary state construction and found the propagators to agree. In this way, it should be possible to construct the propagator for more complex systems, provided one can relate the desired system to a simpler one for which the propagator is known. For example, the standard (singletwist) Klein Bottle is a mix between the two systems solved here, and as such its propagator would be a blend between the torus propagator in one dimension and the non-standard Klein Bottle in the other. One can treat composite systems in much the same way, eg: the torus is composed of two circles, one in each dimension x and y. Furthermore, we have also treated briefly the topic of applying further constraints to these systems. As a result, we have shown that not all constraints are viable candidates to be applied, they must commute with the symmetry operators of the system. On the torus, the constraint py ψ = 0 does commute with the symmetries, and the resulting constrained states resemble those of the circle as expected. On the Klein Bottle, the momentum 2 operator does not commute with one of the symmetries (namely, SxK ), and so the constraint cannot be applied. Attempting to do so exposed the fact that the eigenstates of the Hamiltonian are not eigenstates of the momentum operator.  26  Bibliography [1] T. M. Apostol. Mathematical Analysis. Addison-Wesley, 2nd edition, 1974. [2] J. S. Dowker. Quantum mechanics and field theory on multiply connected and on homogeneous spaces. J. Phys. A: Gen. Phys., 5(936), 1972. [3] R. Friedberg et al. A soluble gauge model with gribov-type copies. Annals Phys., 246(381), 1996. [4] M. G. G. Laidlaw and C. M. DeWitt. Feynman functional integrals for systems of indistinguishable particles. Phys. Rev. D, 3(1375), 1971. [5] L. S. Schulman. A path integral for spin. Phys. Rev., 176(1558), 1968. [6] L. S. Schulman. Techniques and Applications of Path Integration. Wiley, 1981.  27  Appendix A  Jacobi Theta Function Identities Beginning with the definition of the Jacobi theta function (2.9): ∞  exp(iπn2 T + 2inZ)  Θ3 (Z, T ) =  (A.1)  n=−∞  we first prove the periodicity identity (2.10), where for (a, b) ∈ Z: exp(iπn2 T + 2in(Z + aπ + bπT ))  Θ3 (Z + aπ + bπT, T ) = n  exp(iπn2 T + 2inZ + 2inbπT )exp(2πina)  = n  (A.2) The second factor in the sum is unity for all values of n and a, and completing the square on b yields: exp(iπn2 T + 2inZ + 2inbπT + iπb2 T − iπb2 T )  Θ3 (Z + aπ + bπT, T ) = n  exp(iπ(n + b)2 T + 2inZ − iπb2 T )  =  (A.3)  n  Finally, we re-index the sum from n to n+b, continuing to write it as being over n (since it goes from -∞ to ∞):  28  Appendix A. Jacobi Theta Function Identities  exp(iπn2 T + 2i(n − b)Z − iπb2 T )  Θ3 (Z + aπ + bπT, T ) = n  exp(iπn2 T + 2inZ)exp(−2ibZ − iπb2 T )  = n  = Θ3 (Z, T )exp(−2ibZ − iπb2 T )  (A.4)  which gives us the identity (2.10). Proving the second required identity for the Jacobi theta function (2.11), we begin by applying the Poisson summation formula to the theta function in a more general form than presented by Apostol [1], for which only the Z=0 case is proven: ∞  ∞ iπn2 T +2inZ  e  Θ3 (Z, T ) =  ∞  2 T +2ixZ  eiπx  =  e2πixn dx  (A.5)  n=−∞ −∞  n=−∞  Recognizing the integral on the right as the Fourier transform of the theta function, we find: ∞  Θ3 (Z, T ) =  (−iT )−1/2 exp(−i n=−∞  (Z + πn)2 ) πT  ∞ −1/2 −iZ 2 /πT  = (−iT )  e  exp(− n=−∞  2iZn iπn2 − ) T T  (A.6)  which, using Θ3 (−Z, T ) = Θ3 (Z, T ), gives us the second identity (2.11): Θ3 (Z, T ) = (−iT )−1/2 exp(Z 2 /iπT )Θ3 (Z/T, −1/T )  (A.7)  29  

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