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

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:

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

The result then follows from this consideration: the association $\alpha\mapsto K^{\alpha}$ of a partial amplitude $K^{\alpha}$ to each element $\alpha$ of the group $\pi$ depends on the injection $f_{ab}$, which in turn, depends on the choice of the mesh function $C$$(x)$. The (absolute value) of the propagator $K$, however, must be the same independently of the adopted mesh function.

The best place to find the details of the proof is still the paper "Feynman Functional Integrals for Systems of Indistinguishable Particles" (1971) by Laidlaw and DeWitt.

Additionally, there is another way to motivate the formula of the propagator $K$ as a sum over partial amplitudes associated to homotopy classes, which is based in a covering space of $X$. This was in fact one of the original reasonings employed by Schulman in "A Path Integral for Spin" (1968) in order to discuss the spin of a (quantum) non-relativistic particle using exclusively the method of path integration.

Roughly, it is like this. Let $\mathbf{X}$ be covering space of $X$ and $\mathrm{p}:\mathbf{X}\longrightarrow X$ the covering projection. Moreover, let $\mathcal{L}$ be the Lagrangian of our system in $X$, for which $S=\int\mathcal{L}dt$, $\mathbf{L}$ the lift of $\mathcal{L}$ to our covering space $\mathbf{X}$ induced by the projection $\mathrm{p}$ and $\mathbf{S}=\int\mathbf{L}dt$ the action on $\mathbf{X}$. To each pair $(a,b)\in X\times X$, choose some $\mathbf{a}\in\mathrm{p}^{-1}(a)$ and let $\mathbf{b}_{\alpha}\in\mathrm{p}^{-1}(b)$ define a sequence in $\mathbf{X}$ indexed by the elements of the fundamental group, that is, $\alpha\in\pi$.

Since the covering space is simply-connected, to each $\alpha\in\pi$, the propagator $K^{\alpha}$ associated to the amplitude for going from $\mathbf{a}$ to $\mathbf{b}_{\alpha}$ in the interval $[t_{a},t_{b}]$ on $\mathbf{X}$ is given, as familiar, by the path integral

$$ K^{\alpha}=\int_{\mathbf{a}}^{\mathbf{b}_{\alpha}}e^{i\mathbf{S}[\mathbf{x}(t)]}\mathcal{D}[\mathbf{x}(t)], $$

where in this case, the functional integral runs over the paths $\mathbf{x}(t):[t_{a},t_{b}]\longrightarrow\mathbf{X}$ connecting $\mathbf{a}$ to $\mathbf{b}_{\alpha}$.

Finally, by the principle of the sum over the alternatives, the propagator $K$ for the amplitude of going from $a$ to $b$ in the time interval $[t_{a},t_{b}]$ on our multiply-connected space $X$ is seem to be the sum over the alternatives for going from $\mathbf{a}$ to $\mathbf{b}_{\alpha}$ for all $\alpha\in\pi$ in the covering $\mathbf{X}$. In effect, we shall again obtain $K=\sum_{\alpha\in\pi}\xi(\alpha)K^{\alpha}$ for some phase factors $\xi(\alpha)\in\mathbb{C}$.

A proof of the result found by Schulman, Laidlaw and DeWitt described above, using the latter approach of covering spaces, may be found in the paper "Quantum mechanics and field theory on multiply connected and on homogenous spaces" (1972) by Dowker. I believe that the best sources for learning the subject are still the original papers cited above (and which I may send upon request). Additionally, if you have access to an university library, it is opportune to give a look at the Chapter 8 of "Functional Integration: Action and Symmetries" by Cartier and DeWitt.