Multitime propagators and the consistency condition

Abstract

For a nonrelativistic quantum system of $N$ particles, the wave function is a function of $3N$ spatial coordinates and one temporal coordinate. The relativistic generalization of this wave function is a function of $N$ time variables known as the multitime wave function, and its evolution is described by $N$ Schrödinger equations, one for each time variable. To guarantee the existence of a nontrivial common solution of these $N$ equations, the $N$ Hamiltonians must satisfy a compatibility condition known as the integrability condition. In this work, the integrability condition is expressed in terms of Lagrangians. The time evolution of a wave function with $N$ time variables is derived in Feynman’s picture of quantum mechanics. However, these evolutions are compatible if and only if the $N$ Lagrangians satisfy a certain relation called the consistency condition, which can be expressed in terms of Wilson line. As a consequence of this consistency condition, the evolution of the wave function gives rise to a key feature called the “path-independence” property on the space of time variables. This suggests that one must consider all possible paths not only on the space of dependent variables (spatial variables) but also on the space of independent variables (temporal variables). Geometrically, this consistency condition can be regarded as a zero-curvature condition and the multitime evolutions can be treated as compatible parallel transport processes on the flat space of time variables.

A.

Here, we show the derivation of the transition from $(t_1,t_2)$ to $(\hat t_1,\hat t_2)$, as shown in Fig. 5a:

$$\langle\hat q_1,\hat q_2|\Phi(\hat t_1,\hat t_2)\rangle=\langle\hat q_1,\hat q_2|U'_2U'_1U_2U_1|\Phi(t_1,t_2)\rangle,$$

and

$$\begin{aligned} \, \Phi&(\hat q_1^{},\hat q_2^{},\hat t_1^{},\hat t_2^{})= \iint\langle\hat q_1^{},\hat q_2^{}|U'_2U'_1U_2^{}U_1^{}|q_1^{},q_2^{}\rangle\langle q_2^{},q_1^{}|\Phi(t_1^{},t_2^{})\rangle\,dq_1^{}\,dq_2^{}= \\ &=\iint\langle\hat q_1^{},\hat q_2^{}|U'_2U'_1U_2^{}U_1^{}|q_1^{},q_2^{}\rangle\Phi(q_1^{},q_2^{},t_1^{},t_2^{})\,dq_1^{}\,dq_2^{}= \\ &=\iiiint \langle\hat q_1^{},\hat q_2^{}|U'_2U'_1U_2^{}|\tilde q_1^{},\tilde q_2^{}\rangle \langle\tilde q_2^{},\tilde q_1^{}|U_1^{}|q_1^{},q_2^{}\rangle\Phi(q_1^{},q_2^{},t_1^{},t_2^{})\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}\,= \\ &=\iiiint \langle\hat q_1^{},\hat q_2^{}|U'_2U'_1U_2^{}|\tilde q_1^{},\tilde q_2^{}\rangle \langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle \langle\tilde q_2^{}|q_2^{}\rangle\Phi(q_1^{},q_2^{},t_1^{},t_2^{})\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}\,= \\ &=\iiiint \langle\hat q_1^{},\hat q_2^{}|U'_2U'_1U_2^{}|\tilde q_1^{},\tilde q_2^{}\rangle \langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle\delta(\tilde q_2^{}-q_2^{})\Phi(q_1^{},q_2^{},t_1^{},t_2^{})\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}\,= \\ &=\iiiint\!\!\!\iint \langle\hat q_1^{},\hat q_2^{}|U'_2U'_1|q'_1,q'_2\rangle \langle q'_2,q'_1|U_2^{}|\tilde q_1^{},\tilde q_2^{}\rangle \langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle\times{} \\ &\kern30pt\times\delta(\tilde q_2-q_2)\Phi(q_1,q_2,t_1,t_2)\,dq'_1\,dq'_2\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiiint\!\!\!\iint \langle\hat q_1^{},\hat q_2^{}|U'_2U'_1|q'_1,q'_2\rangle \langle q'_2|U_2^{}|\tilde q_2^{}\rangle \langle q'_1|\tilde q_1^{}\rangle\langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle\times{} \\ &\kern30pt\times\delta(\tilde q_2-q_2)\Phi(q_1,q_2,t_1,t_2)\,dq'_1\,dq'_2\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiiint\!\!\!\iint \langle\hat q_1^{},\hat q_2^{}|U'_2U'_1|q'_1,q'_2\rangle \langle q'_2|U_2^{}|\tilde q_2^{}\rangle\delta(q'_1-\tilde q_1^{})\langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle\times{} \\ &\kern30pt\times\delta(\tilde q_2-q_2)\Phi(q_1,q_2,t_1,t_2)\,dq'_1\,dq'_2\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiiint\!\!\!\iiiint \langle\hat q_1^{},\hat q_2^{}|U'_2|\bar q_1^{},\bar q_2^{}\rangle \langle\bar q_1^{},\bar q_2^{}|U'_1|q'_1,q'_2\rangle\langle q'_2|U_2^{}|\tilde q_2^{}\rangle\times{} \\ &\kern30pt\times\delta(q'_1-\tilde q_1^{})\langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle \delta(\tilde q_2^{}-q_2^{})\Phi(q_1^{},q_2^{},t_1^{},t_2^{}) \,d\bar q_1^{}\,d\bar q_2^{}\,dq'_1\,dq'_2\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiiint\!\!\!\iiiint \langle\hat q_1^{},\hat q_2^{}|U'_2|\bar q_1^{},\bar q_2^{}\rangle \langle\bar q_1^{}|U'_1|q'_1\rangle\langle\bar q_2^{}|q'_2\rangle\langle q'_2|U_2^{}|\tilde q_2^{}\rangle\times{} \\ &\kern30pt\times\delta(q'_1-\tilde q_1^{})\langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle \delta(\tilde q_2^{}-q_2^{})\Phi(q_1^{},q_2^{},t_1^{},t_2^{}) \,d\bar q_1^{}\,d\bar q_2^{}\,dq'_1\,dq'_2\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiiint\!\!\!\iiiint \langle\hat q_1^{},\hat q_2^{}|U'_2|\bar q_1^{},\bar q_2^{}\rangle \langle\bar q_1^{}|U'_1|q'_1\rangle\delta(\bar q_2^{}{-}q'_2)\langle q'_2|U_2^{}|\tilde q_2^{}\rangle\times{} \\ &\kern30pt\times\delta(q'_1-\tilde q_1^{})\langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle \delta(\tilde q_2^{}-q_2^{})\Phi(q_1^{},q_2^{},t_1^{},t_2^{}) \,d\bar q_1^{}\,d\bar q_2^{} \,d\bar q_1^{}\,d\bar q_2^{}\,dq'_1\,dq'_2\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiiint\!\!\!\iiiint \langle\hat q_2^{}|U'_2|\rangle\langle\hat q_1^{}|\bar q_1^{}\rangle \langle\bar q_1^{}|U'_1|q'_1\rangle\delta(\bar q_2^{}{-}q'_2)\langle q'_2|U_2^{}|\tilde q_2^{}\rangle\times{} \\ &\kern30pt\times\delta(q'_1-\tilde q_1^{})\langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle \delta(\tilde q_2^{}-q_2^{})\Phi(q_1^{},q_2^{},t_1^{},t_2^{}) \,d\bar q_1^{}\,d\bar q_2^{}\,dq'_1\,dq'_2\,d\tilde q_1^{}\,d\tilde q_2^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiiint \langle\hat q_2^{}|U'_2|\bar q_2^{}\rangle \int\delta(\hat q_1^{}-\bar q_1^{}\langle\bar q_1^{}|U'_1|q'_1\rangle \,d\bar q_1^{} \int\delta(\bar q_2^{}-q'_2)\langle q'_2|U_2^{}|\tilde q_2^{}\rangle\,dq'_2\times{} \\ &\kern38pt\times\int \delta(q'_1-\tilde q_1^{})\langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle\,dq'_1 \int \delta(\tilde q_2^{}-q_2^{})\Phi(q_1^{},q_2^{},t_1^{},t_2^{})\,d\tilde q_2^{}\,d\bar q_2^{}\,d\tilde q_1^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiiint \langle\hat q_2^{}|U'_2|\bar q_2^{}\rangle \langle\bar q_2^{}|U_2^{}|q_2^{}\rangle \langle\hat q_1^{}|U'_1|\tilde q_1^{}\rangle \langle\tilde q_1^{}|U_1^{}|q_1^{}\rangle\Phi(q_1^{},q_2^{},t_1^{},t_2^{})\,d\bar q_2^{}\,d\tilde q_1^{}\,dq_1^{}\,dq_2^{}= \\ &=\iiint \langle\hat q_2^{}|U'_2(\hat t_2^{}-t'_2)|\bar q_2^{}\rangle\langle\bar q_2^{}|U_2^{}(t'_2-t_2^{})|q_2^{}\rangle\,d\bar q_2^{}\,\times{}\ \\ &\kern38pt\times\int \langle\hat q_1^{}|U'_1(\hat t_1^{}-t'_1)|\tilde q_1^{}\rangle \langle\tilde q_1|U_1(t'_1-t_1^{})|q_1^{}\rangle\,d\tilde q_1^{}\, \Phi(q_1^{},q_2^{},t_1^{},t_2^{})\,dq_1^{}\,dq_2^{}= \\ &=\iiint K_2^{}(\hat q_2^{},\hat t_2^{};\bar q_2^{},t'_2)K_2^{}(\bar q_2^{},t'_2;q_2^{},t_2^{})\,d\bar q_2^{} \times{} \\ &\kern38pt\times\int K_1^{}(\hat q_1^{},\hat t_1^{};\tilde q_1^{},t'_1)K_1^{}(\tilde q_1^{}t'_1;q_1^{},t_1^{})\,d\tilde q_1^{}\, \Phi(q_1,q_2,t_1,t_2)\,dq_1^{}\,dq_2^{}\,. \end{aligned}$$

This proves expression (3.30) for the counter-clockwise transition in a two-time system.