35.3 Quantization and path integrals

35.3 Quantization and path integrals

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After use of the Legendre transform to pass to a Hamiltonian system, one then faces the question of how to construct a corresponding quantum theory. The method of “canonical quantization” is the one we have studied, taking the position coordinates $q_j$ to operators $Q_j$ and momentum coordinates $p_j$ to operators $P_j$ , with $Q_j$ and $P_j$ satisfying the Heisenberg commutation relations. By the Stone von-Neumann theorem, up to unitary equivalence there is only one way to do this and realize these operators on a state space $\mathcal{H}.$ Recall though that the Groenewold-van Hove no-go theorem says that there is an inherent operatorordering ambiguity for operators of higher order than quadratic, thus for such operators providing many diferent possible quantizations of the same classical system (diferent though only by terms proportional to $\hslash )$ . In cases where the Legendre transform is not an isomorphism, a new set of problems appear when one tries to pass to a quantum system since the standard method of canonical quantization will no longer apply, and new methods are needed.

There is however a very diferent approach to relating classical and quantum theories, which completely bypasses the Hamiltonian formalism, just using the Lagrangian. This is the path integral formalism, which is based upon a method for calculating matrix elements of the time evolution operator

$$⟨q_T|e^{-\frac{i}{\hslash }HT}|q_0⟩$$

in the position eigenstate basis in terms of an integral over the space of paths that ${}_{\mathrm{go}}$ from position $q_0$ to position $q_T$ in time $T$ (we will here only treat the $d=1\mathrm{case})$ . Here $|q_0$ is an eigenstate of $Q$ with eigenvalue $q_0$ (a delta-function at $q_0$ in the position space representation), and $|q_T$ has $Q$ eigenvalue $q_T$ . This matrix element has a physical interpretation as the amplitude for a particle starting at $q_0$ at $t=0$ to have position $q_T$ at time $T,$ with its norm-squared giving the probability density for observing the particle at position $q_T$ . It is also the kernel function that allows one to determine the wavefunction $\psi (q,T)$ at any time $t=T$ in terms of its initial value at $t=0$ , by calculating

$$\psi (q_T,T)={\int }_{-\infty }^{\infty }⟨q_T|e^{-\frac{i}{\hslash }HT}|q_0⟩\psi (q_0,0)d{q}_0$$

Note that for the free particle case this is the propagator that we studied in section 12.5.

To try and derive a path-integral expression for this, one breaks up the interval $[0,T]$ into $N$ equal-sized sub-intervals and calculates

$$⟨q_T|(e^{-\frac{i}{N\hslash }HT}{)}^N|q_0⟩$$

If the Hamiltonian is a sum $H=K+V$ , the Trotter product formula shows that

$$⟨q_T|e^{-\frac{i}{\hslash }HT}|q_0⟩=\underset{N\to \infty }{\lim }⟨q_T|(e^{-\frac{i}{N\hslash }KT}{e}^{-\frac{i}{N\hslash }VT}{)}^N|q_0⟩\text{\hspace{1em}(35.2)}$$

If $K(P)$ can be chosen to depend only on the momentum operator $P$ and $V(Q)$ depends only on the operator $Q$ , then one can insert alternate copies of the identity operator in the forms

$${\int }_{-\infty }^{\infty }|q⟩⟨q|dq=1,{\int }_{-\infty }^{\infty }|p⟩⟨p|dp=1$$

This gives a product of terms of the form

$$⟨q_{t_j}|e^{-\frac{i}{N\hslash }K(P)T}|p_{t_j}⟩⟨p_{t_j}|e^{-\frac{i}{N\hslash }V(Q)T}|q_{t_{j-1}}⟩$$

where the index $j$ goes from 0 to $N,t_j=jT/N$ and the variables $q_{t_j}$ and $p_{t_{\mathcal{I}}}$ will be integrated over.

Such a term can be evaluated as

$$\begin{array}{rlr}& & ⟨q_{t_j}|p_{t_j}⟩⟨p_{t_j}|q_{t_{j-1}}⟩e^{-\frac{i}{N\hslash }K(p_{t_j})T}{e}^{-\frac{i}{N\hslash }V(q_{t_{j-1}})T}\\ & & =\frac{1}{\sqrt{2\pi \hslash }}{e}^{\frac{i}{\hslash }{q}_{t_j}{p}_{t_j}}\frac{1}{\sqrt{2\pi \hslash }}{e}^{-\frac{i}{\hslash }{q}_{t_{j-1}}{p}_{t_j}}{e}^{-\frac{i}{N\hslash }K(p_{t_j})T}{e}^{-\frac{i}{N\hslash }V(q_{t_{j-1}})T}\\ & & =\frac{1}{2\pi \hslash }{e}^{\frac{i}{\hslash }{p}_{t_j}(q_{t_j}-q_{t_{j-1}})}{e}^{-\frac{i}{N\hslash }(K(p_{t_j})+V(q_{t_{j-1}}))T}\end{array}$$

The N factors of this kind give an overall factor of $(\frac{1}{2\pi \hslash }{)}^N$ times something which is a discretized approximation to

$$e^{\frac{i}{\hslash }{\int }_0^T(p\dot{q}-h(q(t),p(t)))dt}$$

where the phase in the exponential is just the action. Taking into account the integrations over $q_{t_{\mathcal{I}}}$ and $p_{t_j}$ one should have something like

$$⟨q_T|e^{-\frac{i}{\hslash }HT}|q_0⟩=\underset{N\to \infty }{\lim }(\frac{1}{2\pi \hslash }{)}^N\prod _{j=1}^N{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }d{p}_{t_j}d{q}_{t_j}{e}^{\frac{i}{\hslash }{\int }_0^T(p\dot{q}-h(q(t),p(t)))dt}$$

although one should not do the first and last integrals over $q$ but fix the first value of $q$ to $q_0$ and the last one to $q_T$ . One can try and interpret this sort of integration in the limit as an integral over the space of paths in phase space, thus a “phase space path integral”.

This is an extremely simple and seductive expression, apparently saying that, once the action $S$ is specified, a quantum system is defined just by considering integrals

$$\int D\gamma e^{\frac{i}{\hslash }S[\gamma ]}$$

over paths $\gamma$ in phase space, where $D\gamma$ is some sort of measure on this space of paths. Since the integration just involves factors of dpdq and the exponential just pdq and $h(p,q)$ , this formalism seems to share the same sort of behavior under the infinite dimensional group of canonical transformations (transformations of the phase space preserving the Poisson bracket) as the classical Hamiltonian formalism. It also appears to solve our problem with operator ordering ambiguities, since the efect of products of $P$ and $Q$ operators at various times can be computed by computing path integrals with various $p$ and $q$ factors in the integrand. These integrand factors commute, giving just one way of producing products at equal times of any number of $P$ and $Q$ operators.

Unfortunately, we know from the Groenewold-van Hove theorem that this is too good to be true. This expression cannot give a unitary representation of the full group of canonical transformations, at least not one that is irreducible and restricts to what we want on transformations generated by linear functions $q$ and $p.$ . Another way to see the problem is that a simple argument shows that by canonical transformations any Hamiltonian can be transformed into a free particle Hamiltonian, so all quantum systems would just be free particles in some choice of variables. For the details of these arguments and a careful examination of what goes wrong, see chapter 31 of [77]. One aspect of the problem is that for successive values of $j$ the coordinates $q_{t_j}$ or $p_{t_j}$ have no reason to be close together. This is an integral over “paths” that do not acquire any expected continuity property as $N\to \infty$ , so the answer one gets can depend on the details of the discretization chosen, reintroducing the operator-ordering ambiguity problem.

One can intuitively see that there is something disturbing about such paths, since one is alternately at each time interval switching back and forth between a q-space representation where $q$ has a fixed value and nothing is known about $p,$ and a p space representation where $p$ has a fixed value but nothing is known about $q.$ The “paths” of the limit are objects with little relation to continuous paths in phase space, so while one may be able to define the limit of equation 35.2, it will not necessarily have any of the properties one expects of an integral over continuous paths.

When the Hamiltonian $h$ is quadratic in the momentum $p,$ the $p_{t_j}$ integrals will be Gaussian integrals that can be performed exactly. Equivalently, the kinetic energy part $K$ of the Hamiltonian operator will have a kernel in position space that can be computed exactly (see equation 12.9). As a result, the $p_{t_j}$ integrals can be eliminated, along with the problematic use of alternating $q-$ space and p-space representations. The remaining integrals over the $q_{t_j}$ are then interpreted as a path integral over paths not in phase space, but in position space. One finds, if $\begin{array}{r}\dot{K}=\frac{P^2}{2m}\end{array}$

$$⟨q_T|e^{-\frac{i}{\hslash }HT}|q_0⟩=$$ $$\underset{N\to \infty }{\lim }(\frac{i2\pi \hslash T}{Nm}{)}^{\frac{N}{2}}\sqrt{\frac{m}{i2\pi \hslash T}}\prod _{j=1}^N{\int }_{-\infty }^{\infty }d{q}_{t_j}{e}^{\frac{i}{\hslash }{\sum }_{j=1}^N\left(\frac{m(q_{t_j}-q_{t_{j-1}}{)}^2}{2T/N}-V(q_{t_j})\frac{T}{N}\right)}$$

In the limit $N\to \infty$ the phase of the exponential becomes

$$S(\gamma )={\int }_0^{T}dt(\frac{1}{2}m({\dot{q}}^2)-V(q(t)))$$

One can try and properly normalize things so that this limit becomes an integral

$$\int D\gamma e^{\frac{i}{\hslash }S[\gamma ]}\text{\hspace{1em}(35.3)}$$

where now the paths $\gamma (t)$ are paths in the position space.

An especially attractive aspect of this expression is that it provides a simple understanding of how classical behavior emerges in the classical limit as $\hslash 0$ The stationary phase approximation method for oscillatory integrals says that, for a function $f$ with a single critical point at $x=x_c(i.e.,f^{\prime }(x_c)=0)$ and for a small parameter $ϵ$, one has

$$\frac{1}{\sqrt{i2\pi ϵ}}{\int }_{-\infty }^{+\infty }dx{e}^{if/ϵ}=\frac{1}{\sqrt{f^{\prime \prime }(c)}}{e}^{if(x_c)/ϵ}(1+O(ϵ))$$

Using the same principle for the infinite dimensional path integral, with $f=S$ the action functional on paths, and $ϵ=\hslash$ , one finds that for $\hslash 0$ the path integral will simplify to something that just depends on the classical trajectory, since by the principle of least action, this is the critical point of S.

Such position-space path integrals do not have the problems of principle of phase space path integrals coming from the Groenewold-van Hove theorem, but they still have serious analytical problems since they involve an attempt to integrate a wildly oscillating phase over an infinite dimensional space. Away from the limit $\hslash 0$ , it is not clear that whatever results one gets will be independent of the details of how one takes the limit to define the infinite dimensional integral, or that one will naturally get a unitary result for the time evolution operator.

One method for making path integrals better defined is an analytic continuation in the time variable, as discussed in section 12.5 for the case of a free particle. In such a free particle case, replacing the use of equation 12.9 by equation 12.8 in the definition of the position space path integral, one finds that this leads to a well-defined measure on paths, Wiener measure. More generally, Wiener measure techniques can be used to define the path integral when the potential energy is non-zero, getting results that ultimately need to be analytically continued back to the physical time variable.

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