{"http:\/\/dx.doi.org\/10.14288\/1.0070863":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Science, Faculty of","type":"literal","lang":"en"},{"value":"Physics and Astronomy, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Scholte-van de Vorst, Matthew","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2009-08-31T21:04:38Z","type":"literal","lang":"en"},{"value":"2009","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Science - MSc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"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.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/12635?expand=metadata","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/extent":[{"value":"234478 bytes","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/elements\/1.1\/format":[{"value":"application\/pdf","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"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\u00a9 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 . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Circle Stationary State Propagator . . . . . . . . . . . . . . . 3 2.2 Circle Symmetry Propagator . . . . . . . . . . . . . . . . . . 5 3 Torus and Klein Bottle Foundations . . . . . . . . . . . . . . 7 3.1 Torus Foundations . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Klein Bottle Foundations . . . . . . . . . . . . . . . . . . . . 8 4 Torus Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.1 Torus Symmetry Propagator . . . . . . . . . . . . . . . . . . 12 4.2 Torus Stationary State Propagator . . . . . . . . . . . . . . . 13 5 Klein Bottle Propagator . . . . . . . . . . . . . . . . . . . . . . 15 5.1 Klein Bottle Symmetry Propagator . . . . . . . . . . . . . . . 15 5.2 Klein Bottle Stationary State Propagator . . . . . . . . . . . 18 6 Additional Constraints . . . . . . . . . . . . . . . . . . . . . . . 23 6.1 Constraint Applied to Torus . . . . . . . . . . . . . . . . . . . 23 6.2 Constraint Applied to Klein Bottle . . . . . . . . . . . . . . . 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 Torus Construction . . . . . . . . . . . . . . . . . . . . . . . . 7 3.2 Klein Bottle Construction . . . . . . . . . . . . . . . . . . . . 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 \u03c8(x0, t0) at some initial point (x0, t0), then the state at some later point (x, t) is given by: \u03c8(x, t) = G(x, t;x0, t0)\u03c8(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: Schul- man [6] derives the propagator for a free particle with time independent Hamiltonian (which allows us to set t0 = 0) to be: G(x, t;x0, 0) = lim N\u2192\u221e ( m 2pii~\u000f )(N+1)\/2 \u222b dx1 . . . dxNexp[ im 2~\u000f N\u2211 j=0 (xj+1 \u2212 xj)2] (1.2) where xN+1 = x and \u000f = t\/(N + 1). This can be simplified to the closed form: G(x, t;x0, 0) = \u221a m 2pii~t exp[ im 2~t (x\u2212 x0)2] (1.3) where the phase is the classical action divided by ~. Determining the propagator becomes more complicated on multiply con- nected spaces, as it is possible to generate two paths having the same end- points which cannot be continuously deformed into each other. In this situa- tion, 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 (\u03c8n(x), En) is a pair of stationary states and ener- 1 Chapter 1. Introduction gies satisfying the Hamiltonian for the space, then the propagator can be expressed as: G(xf , t;xi, 0) = \u221e\u2211 n=\u2212\u221e \u03c8n(xf )\u03c8\u2217n(xi)exp(\u2212iEnt) (1.4) 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 ob- tain 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 M from a simply connected covering space M\u0303 , the problem becomes much simpler. Paths in M from m to n (m,n \u2208M) correspond to paths in M\u0303 from some fixed m\u0303 \u2208 p\u22121(m) to each n\u0303i \u2208 p\u22121(n), where i runs over all pre-images of n. Then we carry out the path integral in M\u0303 (using the projection p to 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 sim- ply 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 our- selves 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 S1 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 = 2pi on the interval [0, 2pi]. Alternatively, the circle is constructed via the identification x = x+ 2pi 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): \u22121 2 \u22022x\u03c8(x) = E\u03c8(x) (2.1) The condition that the system is on the circle is enforced by requiring the state \u03c8(x) also obey the boundary condition: SS 1 \u03c8(x) = \u03c8(x+ 2pi) (2.2) where SS 1 is a unitary symmetry operator which commutes with the Hamil- tonian, [SS 1 , H] = 0. Hence, \u03c8(x) can be simultaneously diagonalized with respect to both operators. If we write S\u03c8 = c\u03c8 (where superscripts and arguments have been sup- pressed), then since: Sn\u03c8(x) = \u03c8(x+ 2npi) (2.3) and: SnSm = Sn+m (2.4) it follows that c = ei\u03c3 (for \u03c3 a real constant) and Bloch\u2019s theorem states that: 3 Chapter 2. Circle Propagator \u03c8(x) = exp( i\u03c3x 2pi )u(x) (2.5) where u(x) = u(x+2pi) is a periodic function. The solution of Schrodinger\u2019s equation (2.1) for \u03c8(x) of this form yields the complete set of eigenstates: \u03c8S 1 n (x) = 1\u221a 2pi exp(inx+ i\u03c3x 2pi ) (2.6) ES 1 n = 1 2 (n+ \u03c3 2pi )2 (2.7) Thus, instead of a unique quantization, one obtains a set of inequivalent quantizations parameterized by \u03c3, and perfect periodicity corresponds to the unique quantization \u03c3 = 0. As a result, stationary state construction of the propagator (1.4) takes the form: GS 1 (xf , t;xi, 0) = \u2211 n 1 2pi exp(inx+ i\u03c3x 2pi \u2212 it 2 (n+ \u03c3 2pi )2) (2.8) where x = xf \u2212 xi, and the propagator argument shall henceforth be sup- pressed. In order to facilitate comparison with the propagator as derived from symmetry considerations, it is useful to introduce the Jacobi theta function defined by: \u03983(Z, T ) = \u221e\u2211 n=\u2212\u221e exp(ipin2T + 2inZ) (2.9) As a trivial result of the definition, \u03983(\u2212Z, T ) = \u03983(Z, T ). Furthermore, two useful identities of the theta function are: \u03983(Z + api + bpiT, T ) = exp(\u2212ipib2T \u2212 2ibZ)\u03983(Z, T ) (2.10) \u03983(Z, T ) = (\u2212iT )\u22121\/2exp(Z2\/ipiT )\u03983(Z\/T,\u22121\/T ) (2.11) for (a, b) \u2208 Z, the proof of which can be found in Appendix A. Substituting the theta function (2.9) yields: GS 1 = 1 2pi exp(i( \u03c3x 2pi \u2212 \u03c3 2t 8pi2 ))\u03983( x 2 \u2212 \u03c3t 4pi ,\u2212 t 2pi ) (2.12) 4 Chapter 2. Circle Propagator which, if we apply the Jacobi identity (2.11) becomes: GS 1 = 1 2pi (i t 2pi )\u22121\/2exp(i( \u03c3x 2pi \u2212 \u03c3 2t 8pi2 ))\u2217 exp(( x 2 \u2212 \u03c3t 4pi )2\/ipi(\u2212 t 2pi ))\u03983(( x 2 \u2212 \u03c3t 4pi )\/(\u2212 t 2pi ), 2pi t ) =( 1 2piit )1\/2exp( ix2 2t )\u03983( \u03c3 2 \u2212 pix t , 2pi t ) (2.13) 2.2 Circle Symmetry Propagator As a comparison with the stationary state construction, we can also use sym- metry 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 num- ber n: GS 1 (xf , t;xi, 0) = \u221e\u2211 n=\u2212\u221e AnG S1 n (xf , t;xi, 0) (2.14) for some non-trivial set of An\u2019s. By varying one endpoint (say xf ) through a full circle, we find that the full propagator is unchanged up to a phase factor, while we have translated GS 1 n \u2192 GS 1 n\u22121. This, combined with (2.14), tells us that: An+1 = ei\u03c3An (2.15) Furthermore, we know that \u2016A0\u2016 = 1, and so fixing the arbitrary phase of A0 to 0 yields: GS 1 = \u2211 n ein\u03c3GS 1 n (2.16) where the argument of the propagator shall again be suppressed. Since one can (locally) define a smooth mapping f : < \u2192 S1 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 for GS 1 n into (2.16) yields: GS 1 = \u2211 n ( 1 2piit )1\/2exp(in\u03c3 + i(x\u2212 2pin)2 2t ) (2.17) 5 Chapter 2. Circle Propagator Substituting the theta function (2.9) into our expression for the propa- gator on the circle (2.17) yields: GS 1 = ( 1 2piit )1\/2exp( ix2 2t )\u03983( \u03c3 2 \u2212 pix t , 2pi t ) (2.18) in agreement with the stationary state propagator (2.13). 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\u2019s equation for a free particle is just: 7 Chapter 3. Torus and Klein Bottle Foundations \u22121 2 (\u22022x + \u2202 2 y)\u03c8 = E\u03c8 (3.2) Eigenstates (non-normalized) and corresponding eigenvalues obeying the boundary conditions (3.1) are: \u03c8T 2 rs = exp(2pii(rx\/b+ sy\/a)) (3.3) ET 2 rs = 1 2 (( 2pir b )2 + ( 2pis a )2) (3.4) for (r, s) \u2208 Z, where \u03c8T 2rs (x, y) = \u03c8T 2 rs (x+ nb, y +ma) for (n,m) \u2208 Z. However, as seen on the circle, perfect periodicity corresponds to the specific quantization in which \u03c3 = 0. The more general quantization is implemented by unitary symmetry operators generated by the fundamental region: ST 2 x f(x, y) = f(x+ b, y) ST 2 y f(x, y) = f(x, y + a) (3.5) Simultaneous normalized eigenstates and eigenvalues of H, ST 2 x , and S T 2 y are then: \u03c8T 2 rs = 1\u221a ab exp(2pii(rx\/b+ sy\/a) + i\u03c3x b + i\u03b4y a ) (3.6) ET 2 rs = 1 2 (( 2pir + \u03c3 b )2 + ( 2pis+ \u03b4 a )2) (3.7) for (r, s) \u2208 Z, where \u03c3 and \u03b4 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 K2 we intro- duce 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) = (\u2212x, y + a) (x, y) = (x+ b,\u2212y) (3.8) Locally, Schrodinger\u2019s 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. \u03c8K 2 kl E K2 kl k parity l parity cos(pikxb )cos( pily a ) pi2 2 (( k b ) 2 + ( la) 2) even even sin(pikxb )cos( pily a ) pi2 2 (( k b ) 2 + ( la) 2) even odd cos(pikxb )sin( pily a ) pi2 2 (( k b ) 2 + ( la) 2) odd even sin(pikxb )sin( pily a ) pi2 2 (( k b ) 2 + ( la) 2) odd 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 \u03c8T 2 kl = exp(pii(kx\/b+ ly\/a)) (3.9) ET 2 kl = pi2 2 (( k b )2 + ( l a )2) (3.10) 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 + (\u22121)k evaluates to 0 if k is odd, and 2 if k is even. Applying this, the states can be written as: \u03c8K 2 kl = sin( pikx b + pi 4 (1 + (\u22121)l))sin(pily a + pi 4 (1 + (\u22121)k)) (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: SK 2 x f(x, y) = f(x+ b,\u2212y) SK 2 y f(x, y) = f(\u2212x, y + a) (3.12) Simultaneous normalized eigenstates and eigenvalues of H, SK 2 x , and SK 2 y are then: \u03c8K 2 kl = \u221a 4 ab sin( pikx b + \u03c3x 2b + pi 4 (1 + (\u22121)l))\u2217 sin( pily a + \u03b4y 2a + pi 4 (1 + (\u22121)k)) (3.13) EK 2 kl = 1 2 (( pik b + \u03c3 2b )2 + ( pil a + \u03b4 2a )2) (3.14) 10 Chapter 3. Torus and Klein Bottle Foundations for (k, l) \u2208 Z, where \u03c3 and \u03b4 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\u2019s motion: as with the circle it is possible to construct this propagator by a purely symmetry- based 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 = S1 \u00d7 S1. 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: GT 2 = \u2211 r,s ArBsG T 2 rs (4.1) where the coefficients, as for the circle, become phase factors and the indi- vidual propagators those of a free particle. Since we are now dealing with the general period b, we must rescale the coordinate by 2pib 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: \u03c8S 1 n = 1\u221a b exp(i 2pi b nx+ i\u03c3x b ) (4.2) ES 1 n = 1 2 ( 2pin b + \u03c3 b )2 (4.3) GS 1 = ( 1 2piit )1\/2exp( ix2 2t )\u03983( \u03c3 2 \u2212 xb 2t , b2 2pit ) (4.4) 12 Chapter 4. Torus Propagator Making these substitutions into the torus propagator (4.1): GT 2 = \u2211 r,s 1 2piit exp( i 2t (x2 + y2) + ir(\u03c3 \u2212 xb t ) + is(\u03b4 \u2212 ya t ) + ib2r2 2t + ia2s2 2t ) (4.5) which, when we substitute the theta function (2.9), becomes: GT 2 = 1 2piit exp( i 2t (x2 + y2))\u03983( \u03c3 2 \u2212 xb 2t , b2 2pit )\u03983( \u03b4 2 \u2212 ya 2t , a2 2pit ) (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: G(xf , yf , t;xi, yi, 0) = \u221e\u2211 r,s=\u2212\u221e \u03c8rs(xf , yf )\u03c8\u2217rs(xi, yi)exp(\u2212iErst) (4.7) and substituting the normalized states (3.6) and energies (3.7), we find that the propagator for the torus is: GT 2 = \u2211 r,s 1 ab exp(2pii(rx\/b+ sy\/a) + i\u03c3x b + i\u03b4y a \u2212 it 2 (( 2pir + \u03c3 b )2 + ( 2pis+ \u03b4 a )2)) (4.8) where x = xf \u2212 xi, y = yf \u2212 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 GT 2 = 1 ab exp( i\u03c3x b + i\u03b4y a \u2212 it\u03c3 2 2b2 \u2212 it\u03b4 2 2a2 )\u2217 \u03983( pix b \u2212 pi\u03c3t b2 ,\u22122pit b2 )\u03983( piy a \u2212 pi\u03b4t a2 ,\u22122pit a2 ) (4.9) Applying the Jacobi identity (2.11), we find: GT 2 = 1 ab ( 2piit b2 )\u22121\/2( 2piit a2 )\u22121\/2exp( i\u03c3x b + i\u03b4y a \u2212 it\u03c3 2 2b2 \u2212 it\u03b4 2 2a2 )\u2217 exp(( pix b \u2212 pi\u03c3t b2 )2\/\u2212 2pi 2it b2 )exp(( piy a \u2212 pi\u03b4t a2 )2\/\u2212 2pi 2it a2 )\u2217 \u03983(( pix b \u2212 pi\u03c3t b2 )\/\u2212 2pit b2 , b2 2pit )\u03983(( piy a \u2212 pi\u03b4t a2 )\/\u2212 2pit a2 , a2 2pit ) (4.10) which simplifies to: GT 2 = 1 2piit exp( i 2t (x2 + y2))\u03983( \u03c3 2 \u2212 xb 2t , b2 2pit )\u03983( \u03b4 2 \u2212 ya 2t , a2 2pit ) (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 symme- try 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 sym- metry operator looks like a single application of the torus symmetry (3.5) with twice the period: (SK 2 x ) 2f(x, y) = SK 2 x f(x+ b,\u2212y) = 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\u0301T 2 . If we take as an initial guess some combination of G\u0301T 2 and SK 2 , we can check certain properties. First, the propagator should only pick up a phase when the symmetry operator is applied (namely \u03c3\/2, since that would produce the required phase for two applications, ie: x goes to x+2b): SK 2 xf GK 2 (xf , yf , t;xi, yi, 0) = exp(i\u03c3\/2)GK 2 (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: \u03c8K 2 kl (xf , yf , t) = G K2(xf , yf , t;xi, yi, 0)\u03c8K 2 kl (xi, yi, 0) (5.3) In satisfying the first property (5.2), let us successively apply SK 2 to G\u0301T 2 until we are left with a phase factor: 15 Chapter 5. Klein Bottle Propagator SK 2 xf G\u0301T 2 =SK 2 xf \u2211 r,s 1 2piit exp( i 2t ((x\u2212 2rb)2 + (y \u2212 2sa)2) + ir\u03c3 + is\u03b4) = \u2211 r,s 1 2piit exp( i 2t ((x+ b\u2212 2rb)2 + (\u2212y\u0304 \u2212 2sa)2) + ir\u03c3 + is\u03b4) = \u2211 r,s 1 2piit exp( i 2t ((x\u2212 b\u2212 2(r \u2212 1)b)2 + (\u2212y\u0304 \u2212 2sa)2)+ i\u03c3 + i(r \u2212 1)\u03c3 + is\u03b4) =exp(i\u03c3) \u2211 r,s 1 2piit exp( i 2t ((x\u2212 b\u2212 2rb)2 + (\u2212y\u0304 \u2212 2sa)2)+ ir\u03c3 + is\u03b4) (5.4) (SK 2 xf )2G\u0301T 2 =(SK 2 xf )2 \u2211 r,s 1 2piit exp( i 2t ((x\u2212 2rb)2 + (y \u2212 2sa)2) + ir\u03c3 + is\u03b4) = \u2211 r,s 1 2piit exp( i 2t ((x+ 2b\u2212 2rb)2 + (y \u2212 2sa)2) + ir\u03c3 + is\u03b4) = \u2211 r,s 1 2piit exp( i 2t ((x\u2212 2(r \u2212 1)b)2 + (y \u2212 2sa)2)+ i\u03c3 + i(r \u2212 1)\u03c3 + is\u03b4) =exp(i\u03c3) \u2211 r,s 1 2piit exp( i 2t ((x\u2212 2rb)2 + (y \u2212 2sa)2) + ir\u03c3 + is\u03b4) =exp(i\u03c3)G\u0301T 2 (5.5) where y\u0304 = yf + yi. Therefore, as an initial propagator, take: GK 2 =4(G\u0301T 2 + exp(\u2212i\u03c3\/2)SK2xf G\u0301T 2 + exp(\u2212i\u03b4\/2)SK2yf G\u0301T 2+ exp(\u2212i\u03c3\/2)exp(\u2212i\u03b4\/2)SK2xf SK 2 yf G\u0301T 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 SK 2 xf GK 2 =4(SK 2 xf G\u0301T 2 + exp(\u2212i\u03c3\/2)(SK2xf )2G\u0301T 2 + exp(\u2212i\u03b4\/2)SK2xf SK 2 yf G\u0301T 2+ exp(\u2212i\u03c3\/2)exp(\u2212i\u03b4\/2)(SK2xf )2SK 2 yf G\u0301T 2) =4(SK 2 xf G\u0301T 2 + exp(i\u03c3\/2)G\u0301T 2 + exp(\u2212i\u03b4\/2)SK2xf SK 2 yf G\u0301T 2+ exp(i\u03c3\/2)exp(\u2212i\u03b4\/2)SK2yf G\u0301T 2) =exp(i\u03c3\/2)GK 2 (5.7) Substituting for the torus propagator (4.6) and applying the symmetry operators, the Klein Bottle propagator becomes: GK 2 = 2 piit (exp( i 2t (x2 + y2))\u03983( \u03c3 2 \u2212 xb 2t , b2 2pit )\u03983( \u03b4 2 \u2212 ya 2t , a2 2pit )+ exp( i 2t ((x+ b)2 + (\u2212y\u0304)2)\u2212 i\u03c3 2 )\u2217 \u03983( \u03c3 2 \u2212 (x+ b)b 2t , b2 2pit )\u03983( \u03b4 2 + y\u0304a 2t , a2 2pit )+ exp( i 2t ((\u2212x\u0304)2 + (y + a)2)\u2212 i\u03b4 2 )\u2217 \u03983( \u03c3 2 + x\u0304b 2t , b2 2pit )\u03983( \u03b4 2 \u2212 (y + a)a 2t , a2 2pit )+ exp( i 2t ((\u2212x\u0304+ b)2 + (\u2212y\u0304 + a)2)\u2212 i\u03c3 2 \u2212 i\u03b4 2 )\u2217 \u03983( \u03c3 2 + (x\u0304\u2212 b)b 2t , b2 2pit )\u03983( \u03b4 2 + (y\u0304 \u2212 a)a 2t , a2 2pit )) (5.8) Applying the boundary conditions (3.8) associated with the symmetry operators to the final endpoints, we can transform x+ b = (xf + b)\u2212 xi \u2212y\u0304 = \u2212yf \u2212 yi } \u2192 { xf \u2212 xi = x yf \u2212 yi = y (5.9) Applying this transformation to individual terms in the propagator (5.8) and factoring yields: 17 Chapter 5. Klein Bottle Propagator GK 2 = 2 piit (1 + exp( \u2212i\u03c3 2 ) + exp( \u2212i\u03b4 2 ) + exp(\u2212 i\u03c3 2 \u2212 i\u03b4 2 ))\u2217 exp( i 2t (x2 + y2))\u03983( \u03c3 2 \u2212 xb 2t , b2 2pit )\u03983( \u03b4 2 \u2212 ya 2t , a2 2pit ) (5.10) 5.2 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: GK 2 = \u2211 k,l 4 ab sin( pikxf b + \u03c3xf 2b + pi 4 (1 + (\u22121)l))\u2217 sin( pikxi b + \u03c3xi 2b + pi 4 (1 + (\u22121)l)) \u2217 ((Y )) \u2217 exp(\u2212iEK2kl t) (5.11) where ((Y)) denotes the equivalent function of x for y (y replaces x, l replaces k, \u03b4 replaces \u03c3, etc.). Applying the product formula for sine, we have: GK 2 = \u2211 k,l 1 ab (cos( pikx\u0304 b + \u03c3x\u0304 2b + pi 2 (1 + (\u22121)l))\u2212 cos( pikx b + \u03c3x 2b )) \u2217 ((Y )) \u2217 exp(\u2212iEK2kl t) (5.12) Rewriting as exponentials via Euler\u2019s formula, and using: exp(i pi 2 (1 + (\u22121)l)) = exp(ipi(l + 1)) (5.13) the propagator becomes: 18 Chapter 5. Klein Bottle Propagator GK 2 = \u2211 k,l 1 4ab ((exp(i(pik + \u03c3 2 ) x\u0304 b + ipi(l + 1))+ exp(\u2212i(pik + \u03c3 2 ) x\u0304 b + ipi(l + 1)))\u2212 (exp(i(pik + \u03c3 2 ) x b ) + exp(\u2212i(pik + \u03c3 2 ) x b )))\u2217 ((Y )) \u2217 exp(\u2212iEK2kl t) (5.14) Applying exp(ipi) = \u22121 and cancelling the resulting overall -1 factor on both ((X)) and ((Y)), with (3.14) substituted the propagator becomes: GK 2 = \u2211 k,l 1 4ab (exp(i(pik + \u03c3 2 ) x\u0304 b + ipil) + exp(\u2212i(pik + \u03c3 2 ) x\u0304 b + ipil)+ exp(i(pik + \u03c3 2 ) x b ) + exp(\u2212i(pik + \u03c3 2 ) x b ))\u2217 exp(\u2212it( \u03c3 2 8b2 + \u03c3pik 2b2 + pi2k2 2b2 )) \u2217 ((Y )) (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: F (u, v, w, z) =exp(i( \u03c3u 2b \u2212 \u03c3 2t 8b2 + \u03b4v 2a \u2212 \u03b4 2t 8a2 ))\u2217 \u03983( \u2212\u03c3pit 4b2 + piw 2b ,\u2212 pit 2b2 ) \u2217\u03983(\u2212\u03b4pit4a2 + piz 2a ,\u2212 pit 2a2 ) (5.16) Substituting this into the propagator yields: 19 Chapter 5. Klein Bottle Propagator GK 2 = 1 4ab (F (x\u0304, y\u0304, x\u0304+ b, y\u0304 + a) + F (x\u0304,\u2212y\u0304, x\u0304+ b,\u2212y\u0304 + a)+ F (\u2212x\u0304, y\u0304,\u2212x\u0304+ b, y\u0304 + a) + F (\u2212x\u0304,\u2212y\u0304,\u2212x\u0304+ b,\u2212y\u0304 + a)+ F (x\u0304, y, x\u0304, y + a) + F (x\u0304,\u2212y, x\u0304,\u2212y + a)+ F (\u2212x\u0304, y,\u2212x\u0304, y + a) + F (\u2212x\u0304,\u2212y,\u2212x\u0304,\u2212y + a)+ F (x, y\u0304, x+ b, y\u0304) + F (x,\u2212y\u0304, x+ b,\u2212y\u0304)+ F (\u2212x, y\u0304,\u2212x+ b, y\u0304) + F (\u2212x,\u2212y\u0304,\u2212x+ b,\u2212y\u0304)+ F (x, y, x, y) + F (x,\u2212y, x,\u2212y)+ F (\u2212x, y,\u2212x, y) + F (\u2212x,\u2212y,\u2212x,\u2212y)) (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)\u2212 xi \u2212y\u0304 = \u2212yf \u2212 yi } \u2192 { xf \u2212 xi = x yf \u2212 yi = y We can also write: x+ b = xf \u2212 (xi \u2212 b) y\u0304 = yf + yi } \u2192 { xf \u2212 xi = x yf \u2212 yi = y (5.18) Applying these transformations to the propagator (5.17) yields: GK 2 = 1 4ab (4F (x\u2212 b, y \u2212 a, x, y) + 2F (x, y \u2212 a, x, y)+ 2F (x,\u2212y \u2212 a, x,\u2212y) + 2F (x\u2212 b, y, x, y)+ 2F (\u2212x\u2212 b, y,\u2212x, y) + F (x, y, x, y)+ F (x,\u2212y, x,\u2212y) + F (\u2212x, y,\u2212x, y) + F (\u2212x,\u2212y,\u2212x,\u2212y)) (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)\u2212 (xi + b) \u2212y = \u2212yf + yi } \u2192 { xf \u2212 xi = x yf \u2212 yi = y (5.20) which allows us to further collapse the propagator into: 20 Chapter 5. Klein Bottle Propagator GK 2 = 1 ab (F (x\u2212 b, y \u2212 a, x, y) + F (x, y \u2212 a, x, y)+ F (x\u2212 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: F (u, v, w, z) =( 2ab ipit )exp(i( \u03c3u 2b \u2212 \u03c3 2t 8b2 + \u03b4v 2a \u2212 \u03b4 2t 8a2 ))\u2217 exp(i( \u2212\u03c3pit 4b2 + piw 2b )2\/ pi2t 2b2 )exp(i( \u2212\u03b4pit 4a2 + piz 2a )2\/ pi2t 2a2 )\u2217 \u03983(( \u2212\u03c3pit 4b2 + piw 2b )\/\u2212 pit 2b2 , 2b2 pit )\u2217 \u03983(( \u2212\u03b4pit 4a2 + piz 2a )\/\u2212 pit 2a2 , 2a2 pit ) (5.22) which simplifies to: F (u, v, w, z) =( 2ab ipit )exp(i( \u03c3(u\u2212 w) 2b + w2 2t + \u03b4(v \u2212 z) 2a + z2 2t ))\u2217 \u03983(( \u03c3 2 \u2212 wb t ), 2b2 pit )\u03983(( \u03b4 2 + za t ), 2a2 pit ) (5.23) Substituting this into our expression for the propagator (5.21) yields: GK 2 = 2 ipit (exp(i( \u2212\u03c3 2 + x2 2t + \u2212\u03b4 2 + y2 2t ))+ exp(i( x2 2t + \u2212\u03b4 2 + y2 2t )) + exp(i( \u2212\u03c3 2 + x2 2t + y2 2t ))+ exp(i( x2 2t + y2 2t )))\u03983(( \u03c3 2 \u2212 xb t ), 2b2 pit )\u03983(( \u03b4 2 + ya t ), 2a2 pit ) (5.24) which can be rewritten as: 21 Chapter 5. Klein Bottle Propagator GK 2 = 2 piit (1 + exp( \u2212i\u03c3 2 ) + exp( \u2212i\u03b4 2 ) + exp(\u2212 i\u03c3 2 \u2212 i\u03b4 2 ))\u2217 exp( i 2t (x2 + y2))\u03983( \u03c3 2 \u2212 xb 2t , b2 2pit )\u03983( \u03b4 2 \u2212 ya 2t , a2 2pit ) (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\u03c8 = 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 = \u2212i\u2202y, 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: py\u03c8 T 2 rs = ( 2pis a + \u03b4 a )\u03c8T 2 rs (6.2) Therefore a momentum operator of the form: pT 2C y = 1 i \u2202y \u2212 \u03b4 a (6.3) will eliminate the constant factor and allow us to satisfy the constraint (6.1) which becomes: pT 2C y \u03c8 T 2 rs = 2pis a \u03c8T 2 rs = 0 (6.4) with the condition s=0. With the new momentum operator, the energy spectrum becomes: ET 2C rs = 1 2 (( 2pir b )2 + ( 2pis a )2) (6.5) and the constrained states and energies become: 23 Chapter 6. Additional Constraints \u03c8T 2C r0 = 1\u221a ab exp( 2piirx b + i\u03c3x b + i\u03b4y a ) (6.6) ET 2C r0 = 1 2 ( 2pir b )2 (6.7) Note that the constrained system matches that of the circle with peri- odicity b, as is reasonable to expect given the nature of the constraint we\u2019ve 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 = \u2212i\u2202y acting on our eigenstates (3.13) yields: py\u03c8 K2 kl =\u2212 i \u221a 4 ab ( pil a + \u03b4 2a )sin( pikx b + \u03c3x 2b + pi 4 (1 + (\u22121)l))\u2217 cos( pily a + \u03b4y 2a + pi 4 (1 + (\u22121)k)) =\u2212 i(pil a + \u03b4 2a )\u03c8\u0303K 2 kl (6.8) so we see that our eigenstates of the Hamiltonian are not eigenstates of the old momentum operator. To see why that is, consider the commutator of py and SK 2 x : [py, SK 2 x ]f(x, y) =(\u2212i\u2202ySK 2 x + S K2 x i\u2202y)f(x, y) =\u2212 i\u2202yf(x+ b,\u2212y) + iSK2x \u2202f(x, y) \u2202y =\u2212 i\u2202f(x+ b,\u2212y) \u2202y \u2212 i\u2202f(x+ b,\u2212y) \u2202y 6= 0 (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 (single- twist) 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 con- straints are viable candidates to be applied, they must commute with the symmetry operators of the system. On the torus, the constraint py\u03c8 = 0 does commute with the symmetries, and the resulting constrained states re- semble those of the circle as expected. On the Klein Bottle, the momentum operator does not commute with one of the symmetries (namely, SK 2 x ), 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 con- nected 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. An- nals 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): \u03983(Z, T ) = \u221e\u2211 n=\u2212\u221e exp(ipin2T + 2inZ) (A.1) we first prove the periodicity identity (2.10), where for (a, b) \u2208 Z: \u03983(Z + api + bpiT, T ) = \u2211 n exp(ipin2T + 2in(Z + api + bpiT )) = \u2211 n exp(ipin2T + 2inZ + 2inbpiT )exp(2piina) (A.2) The second factor in the sum is unity for all values of n and a, and completing the square on b yields: \u03983(Z + api + bpiT, T ) = \u2211 n exp(ipin2T + 2inZ + 2inbpiT + ipib2T \u2212 ipib2T ) = \u2211 n exp(ipi(n+ b)2T + 2inZ \u2212 ipib2T ) (A.3) Finally, we re-index the sum from n to n+b, continuing to write it as being over n (since it goes from -\u221e to \u221e): 28 Appendix A. Jacobi Theta Function Identities \u03983(Z + api + bpiT, T ) = \u2211 n exp(ipin2T + 2i(n\u2212 b)Z \u2212 ipib2T ) = \u2211 n exp(ipin2T + 2inZ)exp(\u22122ibZ \u2212 ipib2T ) = \u03983(Z, T )exp(\u22122ibZ \u2212 ipib2T ) (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: \u03983(Z, T ) = \u221e\u2211 n=\u2212\u221e eipin 2T+2inZ = \u221e\u2211 n=\u2212\u221e \u222b \u221e \u2212\u221e eipix 2T+2ixZe2piixndx (A.5) Recognizing the integral on the right as the Fourier transform of the theta function, we find: \u03983(Z, T ) = \u221e\u2211 n=\u2212\u221e (\u2212iT )\u22121\/2exp(\u2212i(Z + pin) 2 piT ) = (\u2212iT )\u22121\/2e\u2212iZ2\/piT \u221e\u2211 n=\u2212\u221e exp(\u22122iZn T \u2212 ipin 2 T ) (A.6) which, using \u03983(\u2212Z, T ) = \u03983(Z, T ), gives us the second identity (2.11): \u03983(Z, T ) = (\u2212iT )\u22121\/2exp(Z2\/ipiT )\u03983(Z\/T,\u22121\/T ) (A.7) 29","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/hasType":[{"value":"Thesis\/Dissertation","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#dateIssued":[{"value":"2009-11","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/isShownAt":[{"value":"10.14288\/1.0070863","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/language":[{"value":"eng","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeDiscipline":[{"value":"Physics","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/provider":[{"value":"Vancouver : University of British Columbia Library","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/publisher":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/rights":[{"value":"Attribution-NonCommercial-NoDerivatives 4.0 International","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#rightsURI":[{"value":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#scholarLevel":[{"value":"Graduate","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/title":[{"value":"Path integrals for multiply connected spaces","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/type":[{"value":"Text","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#identifierURI":[{"value":"http:\/\/hdl.handle.net\/2429\/12635","type":"literal","lang":"en"}]}}