# Why isn't the path integral defined for non homotopic paths?

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Context

In the Aharonov Bohm effect, there is a solenoid which creates a magnetic field. Since the electron cannot be inside the solenoid, the configuration space is not simply connected.

Question

I've read in this paper, that the path integral is defined only for paths in the same homotopy class in the configuration space. But I don't see the reason for this. Could someone explain it or give any reference?

It seems that Laidlaw, DeWitt and Schulman have done some work, but I didn't see any proof. And Feynman & Hibbs don't seem to mention it.

Furthermore, does the same problem arise in standard variational calculus when one applies Hamilton's principle?

This post imported from StackExchange Physics at 2014-11-19 20:06 (UTC), posted by SE-user jinawee
The path integral is defined as a sum over all possible paths. In order to do such a summation, you have to integrate within each homotopy class and then sum up all homotopy classes. This is also what the paper states in Theorem 3.1.

This post imported from StackExchange Physics at 2017-01-08 20:03 (UTC), posted by SE-user Heidar
@Heidar What I've read is that one has to calculate the transition amplitude for each homotopy class and sum them up with their respective weights. But I don't see why this is necessary. Why can't the weight function be one?

This post imported from StackExchange Physics at 2017-01-08 20:03 (UTC), posted by SE-user jinawee
@Heidar What I've read is that one has to calculate the transition amplitude for each homotopy class and sum them up with their respective weights. But I don't see why this is necessary. Why can't the weight function be one?

This post imported from StackExchange Physics at 2014-11-19 20:06 (UTC), posted by SE-user jinawee
Maybe I am misunderstanding this, but if you consider a path integral in a covering space, don't you expect weight function be different? [Update] I found a passage that may help you in p219 of The Global Approach to Quantum Field Theory by B. DeWitt. If I understand him correctly, weight is an ambiguity when we define path integral. In a normal derivation, all weights can be equal, but the derivation assumes the existence of a functional Fourier transform that in turn assumes a vector space structure. So, for topologically non-trivial configuration space, this ambiguity remains.
@Autolatry If you mean that the domain is not simply connected because the function has a singularity, this is not the case. An example is the Aharonov Bohm effect where the electron cannot go through the solenoid, but describes loops around it. So I don't see how you could apply analytic continuation. I'll try to clarify the question when I get home.

This post imported from StackExchange Physics at 2017-01-08 20:03 (UTC), posted by SE-user jinawee
@Autolatry If you mean that the domain is not simply connected because the function has a singularity, this is not the case. An example is the Aharonov Bohm effect where the electron cannot go through the solenoid, but describes loops around it. So I don't see how you could apply analytic continuation. I'll try to clarify the question when I get home.

This post imported from StackExchange Physics at 2014-11-19 20:06 (UTC), posted by SE-user jinawee

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The principle of the superposition of quantum states, or, as I shall refer to it, the sum over the alternatives, holds for particles belonging to a multiply-connected space in the same way as it holds for particles belonging to a simply-connected one, since it is one of the fundamental principles of quantum theory. On the other hand, what must be better explained here is why the sum over the alternatives, or in the present case, over the paths, in a simply-connected space can be constructed as a single path integral, differently from multiply-connected spaces.

First, I shall begin with an intuitive argument. Let $X$ be a "nice" topological space (we mean, for instance, that $X$ is arcwise connected or locally simply connected), $a,b\in X$, $\Omega(a,b)$ the set of paths $[t_{a},t_{b}]\longrightarrow X$ from $a$ to $b$ and $t_{b}>t_{a}>0$. To each $x(t)\in\Omega(a,b)$, we associate an amplitude $\phi[x(t)]$. Recall that, heuristically, we write the following proportionally relation for the propagator $K=K(b,t_{b};a,t_{a})$,

$$K\sim\sum_{x(t)\in\Omega(a,b)}\phi[x(t)].$$

If $S$ is the action governing the dynamics of our system and if $t_{b}-t_{a}$ is small enough, we know that $\phi[x(t)]\sim e^{iS[x(t)]}$ (where we have assumed $h=2\pi$). But there is no a priori reason, without evoking any property of $X$, to ensure that all the paths should contribute to $K$ with the same phase. For example, if $x(t),y(t)\in\Omega(a,b)$, why we cannot have

$$K\sim e^{iS[x(t)]}-e^{iS[y(t)]}+...\,\,\,?$$

It turns out that if our topological space $X$ is simply-connected, we can always deform the path $x(t)$ to $y(t)$ continuously, a deformation which, in effect, should make $\phi[x(t)]$ approach $\phi[y(t)]$ continuously too. Formally, $$\phi[y(t)]=\lim\phi[x(t)]=e^{iS[y(t)]},\,\,\,\mbox{as }x(t)\rightarrow y(t)\mbox{ continuously.}$$ From this, we can conclude two things:

• Paths in a simply connected space contribute to the total amplitude with the same phase. So if $X$ is simply connected, we can then write the familiar expression $$K\sim\sum_{x(t)\in\Omega(a,b)}e^{iS[x(t)]},$$ which, upon introducing the appropriate measure, results in the Feynman path integral $$K=\int_{\Omega(a,b)}e^{iS[x(t)]}\mathcal{D}x(t).$$
• Paths in the same homotopy class contribute to the total amplitude with the same phase. So, for the propagator $K^{q}$ restricted to paths constrained in the homotopy class $q$, we can write similarly $$K^{q}\sim\sum_{x(t)\in q}e^{iS[x(t)]},$$ that also becomes a path integral $$K^{q}=\int_{q}e^{iS[x(t)]}\mathcal{D}x(t),$$ but whose domain of functional integration is now $q$. Each such $K^{q}$ is called a partial amplitude.

Since the principle of the sum over the alternatives allows us to write the propagator $K$ as the sum of the amplitudes of each homotopy class $q$ individually (namely, the partial amplitudes), each one contributing with a phase that will be labeled by $\xi_{q}\in\mathbb{C},|\xi_{q}|=1$, we have that $$K=\sum_{q\in\pi(a,b)}\xi_{q}K^{q},$$ where $\pi(a,b)$ is the set of all homotopy classes for the paths from $a$ to $b$. This answers the question raised by jinawee, I hope.

But now, it will be instructive if we sketch on the proof of that result discovered first by Schulman and proved a little later by Laidlaw and DeWitt. Namely, that the set of phases $\{\xi_{q}\}$ can be "identified" with a scalar unitary representation of the fundamental group of $X$. The idea is the following. Let $c\in X$ fixed and choose $C(x)$ to be any path connecting $c$ to whatever $x\in X$. Such $C(\alpha)$ is known as a homotopy mesh. To each pair $(a,b)\in X\times X$, we can construct a mapping

$$f_{ab}:\pi(c)\longrightarrow\pi(a,b)$$

by $f_{ab}(\alpha)=[C^{-1}(a)]\alpha[C(b)]$. This is an injection between the fundamental group $\pi\equiv\pi(c)$ at $c$ and the homotopy class $\pi(a,b)$, allowing us to label the propagator $K^{q}$ and the phase factor $\xi_{q}$ associated to a homotopy class $q$ with the elements of the fundamental group $\pi$, say,

$$K^{q}\rightarrow K^{\alpha},\,\,\,\xi_{q}\rightarrow\xi(\alpha)$$

iff $f_{ab}(\alpha)=q$. So finally, our propagator assumes the form of a sum over the elements of a group:

$$K=\sum_{\alpha\in\pi}\xi(\alpha)K^{\alpha}.$$