Path integrals in a multiply-connected configuration space (50 years after)

All the quantum evolution operators $\hat{U}(t_{f},t_{i})$ share the following characteristic properties: for any times $(t_{i},t,t_{f})$, we have the composition law

The chronological ordering is completely free, in particular one cannot impose systematically $t\in[t_{f},t_{i}]$ because conditions (4) do not allow to conclude that $\smash{\big{(}\hat{U}(t_{f},t_{i})\big{)}^{*}=\big{(}\hat{U}(t_{f},t_{i})\big{)}^{-1}}$ if $\smash{\hat{U}(t,t)}$ cannot be decomposed into $\smash{\hat{U}(t,t_{I})\hat{U}(t_{I},t)}$ for any $t_{I}$: it is required to use the composition law between operators whose arguments change their chronological order one from the other. We note also that (4b) is not a consequence of (4a) (by taking $t_{f}=t_{i}$ for instance) if we refrain posing a priori that the $\hat{U}$’s are invertible. The propagators $K(q_{f},t_{f},q_{i},t_{i})$ in configuration space $Q$ are (generalised) functions of two points $(q_{f},q_{i})$ in $Q$ that can be thought has the matrix elements $\langle q_{f}|\hat{U}(t_{f},t_{i})|q_{i}\rangle$ where $\{|q\rangle\}$ denotes a non-normalisable basis of the Hilbert space on which the $\hat{U}$’s are defined. The properties (4) have their exact translation in terms of propagators (Pauli, 1957/73, eqs. 30.6,7,8)

We denote by $\mathrm{d}^{\text{{l}}}q$ a given measure on $Q$ (including a non-homogeneous Jacobian when curvilinear coordinates are used) associated with the Dirac function $\delta$ such that $\int_{Q}f(q)\delta(q^{\prime}-q)\,\mathrm{d}^{\text{{l}}}q=f(q^{\prime})$ for any test function $f$ defined on $Q$. It is important to note that the integral that constitutes the right-hand side of (5a) covers the whole $Q$: this is the reason why, when dividing the evolution into a sequence of infinitesimal-time slices, to build up, by their composition, the path integral

the integration domain includes all the paths $\mathscr{C}$ on $Q$ such that $q(t_{i})=q_{i}$ and $q(t_{f})=q_{f}$. Since we want to emphasize the topological properties, we ought not to choose an explicit expression of the action $S$ and actually we will not. We will retain only its additivity by concatenation of paths:

(the latter does not assume any time-reversal invariance, which is in fact not satisfied as soon as a magnetic field is present; rather, it may provide a definition of the transformed Lagrangian under $t\mapsto t_{f}+t_{i}-t$). We will neither use a precise definition of the path integrals. We shall assume that actually the right-hand side of (6) satisfies (5) which not an easy statement to prove or, conversely, that must be included in any constructivist approach. An important departure from the original construction proposed by Feynman and Hibbs (1965, eq. (4-28)) is that their propagator is defined to be zero for $t_{f}<t_{i}$ or, equivalently, they consider the matrix elements of $\hat{U}(t_{f},t_{i})$ multiplied by the Heaviside step function $\Theta(t_{f}-t_{i})$. We will rather not to because we want to preserve the property (5c) which is essential to the group property of the $\hat{U}$’s; we will keep working with a function $K$ that fulfills the same evolution equation as a normalisable state, namely the time dependent Schrödinger equation, without any supplementary $\delta(t-t_{i})$ terms turning it into a retarded Green function. For the same reason, we will not allow to use the Wick-substitution “mantra” that leads to an irreversible evolution governed by a semi-group. The difficulty of defining mathematically an oscillatory path integral is the sign that using an imaginary time is not harmless from the physical point of view; as soon as we suppress, by construction, the central notion of quantum interferences, one is expected to miss a lot of physics including the topological phases as they appear in (1). We will show how the latter are directly connected to the unitarity of the evolution.

For $t_{f}\neq t_{i}$, one expects the function $(q_{f},q_{i})\mapsto K(q_{f},t_{f},q_{i},t_{i})$ to be smooth on the configuration space if the Schrödinger time-dependent equation satisfied by $K$ involves potentials that are regular enough. In particular, for a model invariant under time translations, from a stationary orthonormal eigenbasis $\{|\phi_{\nu}\rangle\}$ labelled by the quantum numbers $\nu$ and corresponding to an energy spectrum $\{\epsilon_{\nu}\}$, we have the following spectral decomposition of $K$ in terms of the corresponding wavefunctions $\phi_{\nu}(q)\smash{\overset{\text{\tiny def}}{=}}\mathinner{\langle{q}|{\phi_{\nu}}\rangle}$ defined on $Q$

and the smoothness of $K$ is at least the same as the smoothness of the stationary wavefunctions.

Once the domain of integration in the right-hand side of (6) has been split in homotopy classes $\mathfrak{c}_{q_{i},q_{f}}$, the weights $E(\mathfrak{c}_{q_{i},q_{f}})$ must be chosen in order for the new $K$ given by (1) to still satisfy (5) and, then, to keep its interpretation of being the density of probability amplitude for reaching the configuration $q_{f}$ at time $t_{f}$ assuming that the system is in the configuration $q_{i}$ at $t_{i}$. On the other hand, the partial path integrals defined by

are not expected to satisfy (5) and, notably, one should expect that

because to obtain the right-hand side requires a path integration with no restriction on the homotopy classes as in (6). In fact, one expects the function $(q_{f},q_{i})\mapsto k_{\mathfrak{c}}(q_{f},t_{f},q_{i},t_{i})$ to have discontinuities in $Q$ because there is no way of deforming continuously, say, a class of paths $\mathfrak{c}_{q_{i},q(s)}$ along a non-trivial loop $q(s)$ without ending with another class of paths. For instance, let us choose $s\in[0,1]\mapsto q(s)$ to be a loop $\mathscr{L}$ in $Q$ with $q(0)=q(1)=q_{f}$ which is not continuously deformable into the constant path $q_{f}$. Then, we can try to maintain the continuity of $s\mapsto k_{\mathfrak{c}}(q(s),t_{f},q_{i},t_{i})$ by transforming continuously the domain of integration of (9) in the following way: to define $\mathfrak{c}_{q_{i},q(s+\mathrm{d}s)}$, all the paths in $\mathfrak{c}_{q_{i},q(s)}$ are concatenated with the portion of $\mathscr{L}$ between $q(s)$ and $q(s+\mathrm{d}s)$ while changing the time parametrisation to keep the paths always defined for $[t_{i},t_{f}]$. Nevertheless, at the end of this process, when the final point $s=1$ is reached, all the paths have been concatenated with $\mathscr{L}$ modulo a time reparametrisation and therefore belong to the class $\mathfrak{c}^{\prime}_{q_{i},q_{f}}=\mathfrak{c}_{q_{i},q_{f}}\cdot\mathfrak{l}_{q_{f},q_{f}}$; the latter is different from the class we started with at $s=0$ if and only if the class $\mathfrak{l}_{q_{f},q_{f}}$ of $\mathscr{L}$ is different from the neutral class $\mathfrak{e}_{q_{f}}$. Having two different integration domains, $k_{\mathfrak{c}}$ and $k_{\mathfrak{c}^{\prime}}$ are a priori different and therefore, even though the limits $s\to 0^{+}$ and $s\to 1^{-}$ lead to the same value $q(0)=q(1)=q_{f}$, they give two different values for $k_{\mathfrak{c}_{q_{i},q(s)}}(q(s),t_{f},q_{i},t_{i})$. Since $\mathscr{L}$ is arbitrary with at least one point of discontinuity on it, we expect to have in $Q$ at least one hypersurface of codimension 1 of points of discontinuity for each $k$.

These singularities of the $k$’s in $Q$ do not contradict the regularity of $(q_{f},q_{i})\mapsto K(q_{f},t_{f},q_{i},t_{i})$ for $t_{f}\neq t_{i}$: within the sum (1), when following the loop $\mathscr{L}$ from $s=0^{+}$ to $s=1^{-}$ a permutation occurs where all terms are swapped one with an other maintaining the continuity of the whole sum. One can then even redefine a continuous family of functions on $Q$ with the help of a Heaviside $\Theta$ functions splitting the two sides of the hypersurfaces of discontinuities of the partial path integrals⁴⁴4If $k_{n}(s_{0}^{+})=k_{\sigma(n)}(s_{0}^{-})\neq k_{n}(s_{0}^{-})$ for a permutation $\sigma$ of the discrete labels $n$, then the function $\tilde{k}_{n}(s)\smash{\overset{\text{\tiny def}}{=}}k_{n}(s)\Theta(s-s_{0})+k_{\sigma(n)}(s)\Theta(s_{0}-s)$ is continuous at $s_{0}$ and one can express the discontinuous functions $k_{n}$ with the continuous functions $\tilde{k}_{n}$: $k_{n}(s)=\tilde{k}_{n}(s)\Theta(s-s_{0})+\tilde{k}_{\sigma^{-1}(n)}(s)\Theta(s_{0}-s)$. but, as we have seen, these functions cannot be associated with the same homotopy class on the two sides of these surfaces.

To keep following the same homotopy class without dealing with cuts, the price to be paid is to somehow establish a distinction between $q(0)$ and $q(1)$ by opening the loop $\mathscr{L}$. This is precisely the reason of using the universal covering space $\bar{Q}$ where the points $\bar{q}$ are identified with the homotopy classes $\mathfrak{c}_{q_{0},q}$ with $q_{0}$ being fixed and $q$ running through $Q$ (the definition of $\bar{Q}$ when attached to $q_{0}$ is given by (3) and its properties are recalled in section 1): the homotopic distinction between paths in $Q$ (sharing the same endpoints) is transferred to a difference in the endpoints in the lifted paths in $\bar{Q}$⁵⁵5The same line of thought leads to the construction of the Riemann surfaces from the complex plane. The latter is unfolded into $n$ connected sheets to avoid the line cuts of the function $z\mapsto z^{1/n}$..

Let us first convert the sum over $\pi_{1}(q_{i},q_{f})$, the class of paths in $Q$ connecting $q_{i}$ to $q_{f}$, to a sum over the fundamental group $\pi_{1}(Q)$ identified with $\pi_{1}(q_{f},q_{f})$. By selecting one $\mathfrak{c}_{0}\in\pi_{1}(q_{i},q_{f})$, (1) reads

Then, pick up one $\bar{q}_{i}\in\Pi^{-1}(q_{i})$, set $q_{0}=q_{f}$ and define $\bar{q}_{f}\smash{\overset{\text{\tiny def}}{=}}\mathfrak{e}_{q_{f}}$. Then each path $\mathscr{C}$ in $Q$ connecting $q_{i}$ to $q_{f}$ is lifted into a unique path $\bar{\mathscr{C}}$ in $\bar{Q}$ connecting $\bar{q}_{i}$ to $g\bar{q}_{f}$ for $g$ in $\pi_{1}(Q)$ associated to $\mathfrak{l}$. When restricted to simply-connected patches on $Q$, all the differential structure of $Q$ can be lifted to $\bar{Q}$, in particular the coordinates charts, the action functional and the measure on paths; the definition of which are part of the translation of the partial path integral into the universal covering space according to

where the integration domain is now the homotopy class $\bar{\mathfrak{c}}_{\bar{q}_{i},g\bar{q}_{f}}$ of the paths $\bar{q}(t)$ in $\bar{Q}$ such that $\bar{q}(t_{i})=\bar{q}_{i}$ and $\bar{q}(t_{f})=g\bar{q}_{f}$. In working in the universal covering space, we have expressed each partial path integrals $k$ into a plain Feynman integral $\bar{K}(g\bar{q}_{f},t_{f},\bar{q}_{i},t_{i})$ where all the paths connecting two points are considered with no restriction on their homotopy classes since, by construction, $\bar{Q}$ is simply connected. The equality (12) concerns the values of $k$ and $\bar{K}$ and not the functions themselves whose arguments are defined in different spaces and whose smoothness properties are not the same.

As any plain Feynman integral, $\bar{K}$ satisfies (5b),

in contrast with (10). Then, if for one reason or another, a linear combination of the form $\sum_{g\in\pi_{1}(Q)}A(g)\bar{K}(g\bar{q}_{f},t_{f},\bar{q}_{i},t_{i})$ vanishes identically for all times, by taking $t_{f}=t_{i}$, this implies that the coefficients $A(g)$ are zero because $g\bar{q}_{f}\neq\bar{q}_{f}$ as soon as $g\neq e$. From (12), this linear independence of the $\delta(g\bar{q}_{f}-\bar{q}_{i})$ for $g\in\pi_{1}(Q)$ is directly transmitted to the $k$’s:

In other words, we have established that the decomposition (1) of a given propagator $K$ is necessarily unique.

For the decomposition (1) to be consistent with (5a), we must have

Take $t_{i}<t_{f}$ choose $t_{I}\in]t_{i},t_{f}[$. Every path $\mathscr{C}$ involved in the integral of the left-hand side, connecting $(q_{i},t_{i})$ to $(q_{f},t_{f})$, is uniquely obtained by concatenation of one path $\mathscr{C}_{2}$ connecting $(q_{I},t_{I})$ to $(q_{f},t_{f})$ to one path $\mathscr{C}_{1}$ connecting $(q_{i},t_{i})$ to $(q_{I},t_{I})$ where $q_{I}$ given by $q(t_{I})$; the class $\mathfrak{c}$ of $\mathscr{C}$ is then uniquely decomposed into $\mathfrak{c}_{1}\cdot\mathfrak{c}_{2}$ where $\mathfrak{c}_{1}$ (resp. $\mathfrak{c}_{2}$) denotes the homotopy class of $\mathscr{C}_{1}$ (resp. $\mathscr{C}_{2}$). Therefore each path of the left-hand side appears once and only once among the paths in the right-hand side.

Conversely, every path involved in the right-hand side is obtained by concatenation of a path connecting $(q_{I},t_{I})$ to $(q_{f},t_{f})$ to a path connecting $(q_{i},t_{i})$ to $(q_{I},t_{I})$ for a given $q_{I}$ and then appears once and only once in the left-hand side.

Moreover, because of the additivity property (7), we can collect the paths of the right-hand side according to

The identification with the left-hand side (15) reads

that is

because of the linear independence of the $k_{\mathfrak{c}}$’s. Then, the generalisation (1) preserves the original Feynman’s interpretation: the probability amplitude brought by the path $\mathscr{C}=\mathscr{C}_{1}\cdot\mathscr{C}_{2}$ to the propagator $K$ remains equal to the product of the amplitudes brought by $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$; since the integral involves all the possible paths, the two concatenated pieces are considered to be independent as soon as the continuity of $\mathscr{C}$ is maintained. This multiplication of the amplitudes reads $E(\mathfrak{c}_{1}\cdot\mathfrak{c}_{2})\,\mathrm{e}^{\frac{\mathrm{i}}{\hbar}S[\mathscr{C}]}=E(\mathfrak{c}_{2})\,E(\mathfrak{c}_{1})\,\mathrm{e}^{\frac{\mathrm{i}}{\hbar}(S[\mathscr{C}_{2}]+S[\mathscr{C}_{1}])}$ which is guaranteed both by (18) and by the additivity of the action with respect to concatenation.

As an immediate consequence of (18) by taking for $\mathfrak{c}_{2}$ any neutral class $\mathfrak{e}_{q}$ and $\mathfrak{c}_{1}\in\pi_{1}(q_{i},q)$ we get

and by choosing $\mathfrak{c}_{2}=\mathfrak{c}_{1}^{-1}$,

The third and last characteristic property of a propagator is the Hermitian conjugation rule (5c). Then we must have

where all the classes $\mathfrak{c}^{-1}$ involved in the right-hand side are made of paths connecting $(q_{f},t_{f})$ to $(q_{i},t_{i})$. Yet, to each of these paths is associated a unique inverse $\mathscr{C}^{-1}$ connecting $(q_{i},t_{i})$ to $(q_{f},t_{f})$ whose action is opposite by virtue of (7): $S[\mathscr{C}^{-1}]=-S[\mathscr{C}]$. On the right-hand side, the sum on the path in $\mathfrak{c}^{-1}$ can be obtained by a sum on the opposite paths in the classes $\mathfrak{c}$: