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path integral

Path integral in quantum mechanics

A simple form of the path integral is realized in quantum mechanics, where it was originally dreamed up by Richard Feynman and then made precise using the Feynman-Kac formula?. Most calculations in practice are still done using perturbation theory.

The Schrödinger equation? says that the rate at which the phase of an energy eigenvector rotates is proportional to its energy:

(1)$$i\hslash \frac{d}{\mathrm{dt}}\psi =H\psi .$$i \hbar \frac{d}{dt} \psi = H \psi.

Therefore, the probability that the system evolves to the final state ${\psi }_F$ after evolving for time $t$ from the initial state ${\psi }_I$ is

(2)$$⟨{\psi }_F\mid e^{-\mathrm{iHt}}\mid {\psi }_I⟩.$$\langle \psi_F|e^{-iHt}|\psi_I\rangle.

Chop this up into time steps $\Delta t=t/N$ and use the fact that

(3)$${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mid q⟩⟨q\mid =1$$\int_{-\infty}^{\infty}|q\rangle\langle q| = 1

to get

(4)$$⟨{\psi }_F\mid e^{-\mathrm{iH}\Delta t}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mid q_{N-1}⟩⟨q_{N-1}\mid {\mathrm{dq}}_{N-1})e^{-\mathrm{iH}\Delta t}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mid q_{N-2}⟩⟨q_{N-2}\mid {\mathrm{dq}}_{N-2})e^{-\mathrm{iH}\Delta t}\cdots e^{-\mathrm{iH}\Delta t}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\mid q_1⟩⟨q_1\mid {\mathrm{dq}}_1)e^{-\mathrm{iH}\Delta t}\mid {\psi }_I⟩$$\langle \psi_F| e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_{N-1} \rangle \langle q_{N-1}| dq_{N-1}\right) e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_{N-2} \rangle \langle q_{N-2}| dq_{N-2}\right) e^{-iH\Delta t} \cdots e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} |q_1 \rangle \langle q_1| dq_1\right) e^{-iH\Delta t} |\psi_I\rangle

(5)$$={\int }_{q_1}\cdots {\int }_{q_{N-2}}{\int }_{q_{N-1}}⟨{\psi }_F\mid e^{-\mathrm{iH}\Delta t}\mid q_{N-1}⟩⟨q_{N-1}\mid e^{-\mathrm{iH}\Delta t}\mid q_{N-2}⟩⟨q_{N-2}\mid e^{-\mathrm{iH}\Delta t}\cdots e^{-\mathrm{iH}\Delta t}\mid q_1⟩⟨q_1\mid e^{-\mathrm{iH}\Delta t}\mid {\psi }_I⟩{\mathrm{dq}}_{N-1}{\mathrm{dq}}_{N-2}\cdots {\mathrm{dq}}_1$$= \int_{q_1} \cdots \int_{q_{N-2}} \int_{q_{N-1}} \langle \psi_F| e^{-iH\Delta t} |q_{N-1} \rangle \langle q_{N-1}| e^{-iH\Delta t} |q_{N-2} \rangle \langle q_{N-2}| e^{-iH\Delta t} \cdots e^{-iH\Delta t} |q_1 \rangle \langle q_1| e^{-iH\Delta t} |\psi_I\rangle dq_{N-1} dq_{N-2} \cdots dq_1

Assume we have the free Hamiltonian $H=p^2/2m.$ Looking at an individual term $⟨q_{n+1}\mid e^{-\mathrm{iH}\Delta t}\mid q_n⟩,$ we can insert a factor of 1 and solve to get

(6)$$\begin{array}{ccc}⟨q_{n+1}\mid e^{-\mathrm{iH}\Delta t}({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{dp}}{2\pi }\mid p⟩⟨p\mid )\mid q_n⟩& =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{dp}}{2\pi }{e}^{-{\mathrm{ip}}^2\Delta t/2m}⟨q_{n+1}\mid p⟩⟨p\mid q_n⟩\\ & =& {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\mathrm{dp}}{2\pi }{e}^{-{\mathrm{ip}}^2\Delta t/2m}{e}^{\mathrm{ip}(q_{n+1}-q_n)}\\ & =& {\left(\frac{-i2\pi m}{\Delta t}\right)}^{\frac{1}{2}}{e}^{i\Delta t(m/2)[(q_{n+1}-q_n)/\Delta t{]}^2}.\end{array}$$\array{\langle q_{n+1}| e^{-iH\Delta t} \left(\int_{-\infty}^{\infty} \frac{dp}{2\pi}|p\rangle \langle p|\right)|q_{n} \rangle &=& \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{-ip^2\Delta t/2m} \langle q_{n+1}|p\rangle \langle p|q_{n} \rangle \\ &=& \int_{-\infty}^{\infty} \frac{dp}{2\pi} e^{-ip^2\Delta t/2m} e^{ip(q_{n+1}-q_n)} \\ &=& \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{1}{2}} e^{i \Delta t (m/2)[(q_{n+1}-q_n)/\Delta t]^2}.}

Defining

(7)$$\int \mathrm{Dq}=\underset{N\to \mathrm{\infty }}{\lim }{\left(\frac{-i2\pi m}{\Delta t}\right)}^{\frac{N}{2}}\prod _{n=0}^{N-1}\int {\mathrm{dq}}_n,$$\int Dq = \lim_{N \to \infty} \left(\frac{-i 2\pi m}{\Delta t}\right)^{\frac{N}{2}} \prod_{n=0}^{N-1} \int dq_n,

and letting $\Delta t\to 0,N\to \mathrm{\infty },$ we get

(8)$$⟨{\psi }_F\mid e^{-\mathrm{iHt}}\mid {\psi }_I⟩=\int \mathrm{Dq}{e}^{i{\int }_0^t\mathrm{dt}\frac{1}{2}m{\dot{q}}^2}.$$\langle \psi_F|e^{-iHt}|\psi_I\rangle = \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2}.

For arbitrary Hamiltonians $H=\frac{p^2}{2m}+V(x),$ we get

(9)$$\begin{array}{ccc}⟨{\psi }_F\mid e^{-\mathrm{iHt}}\mid {\psi }_I⟩& =& \int \mathrm{Dq}{e}^{i{\int }_0^t\mathrm{dt}\frac{1}{2}m{\dot{q}}^2-V(x)}\\ & =& \int \mathrm{Dq}{e}^{i{\int }_0^tℒ(\dot{q},q)\mathrm{dt}}\\ & =& \int \mathrm{Dq}{e}^{\mathrm{iS}(q)},\end{array}$$\array{\langle \psi_F|e^{-iHt}|\psi_I\rangle &=& \int Dq e^{i \int_0^t dt \frac{1}{2}m \dot{q}^2 - V(x)} \\ &=& \int Dq e^{i\int_0^t\mathcal{L}(\dot{q},q) dt} \\ &=& \int Dq e^{iS(q)}, }

where $S(q)$ is the action functional.

Is there an easy way to see how the Hamiltonian transforms into the Lagrangian in the exponent?

References

For simple systems like particles propagating on a Riemannian manifold and charged under a gauge field given by a connection on a bundle the path integral can be made pretty much rigorous

• Christian Baer?, Frank Pfaeffle?, Path integrals on manifolds by finite dimensional approximation (arXiv)

Zoran: usually the QUADRATIC Hamiltonians are the ones for which the integral is well understood in several approaches; and of course many cases corresponding to the exactly solvable models.

The path integral in the bigger picture

Ours is the age whose central fundamental theoretical physics question is:

What is quantum field theory?

A closely related question is:

What is the path integral ?

After its conception by Feynman in the middle of the 20th century It was notably Edward Witten’s achievement in the late 20th century to make clear the vast potential for fundamental physics and pure math underlying the concept of the quantum field theoretic path integral.

And yet, among all the aspects of QFT, the notion of the path integral is the one that has resisted attempts at formalization the most.

While functorial quantum field theory is the formalization of the properties that the locality and the sewing law of the path integral is demanded to have – whatever the path integral is, it is a process that in the end yields a functor on a (infinity,n)-category of cobordisms – by itself, this sheds no light on what that procedure called “path integration” or “path integral quantization” is.

The single major insight into the right higher categorical formalization of the path integral is probably the idea indicated in

• Dan Freed

  • Quantum groups from path integrals (arXiv)

  • Higher algebraic structures and quantization (arXiv)

which says that

• it is wrong to think of the action functional that the path integral integrates over as just a function: it is a higher categorical object;

• accordingly, the path integral is not something that just controls the numbers or linear maps assigned by a $d$-dimensional quantum field theory in dimension $d$: also the assignment to higher codimensions is to be regarded as part of the path integral;

  • notably: the fact that quantum mechanics assigns a (Hilbert) space of sections of a vector bundle to codimension 1 is to be regarded as due to a summing operation in the sense of the path integral, too: the space of sections of a vector bundle is the continuum equivalent of the direct sum of its fibers

More recently, one sees attempts to formalize this observation of Freed’s, notably in the context of the cobordism hypothesis:

• geometric infinity-function theory is used to compute at least something like a path integral in codimension 1 and 2 in the context of sigma-model QFT;

• something not unsimilar is here, the underlying idea of which for a toy example is spelled out at

  exercise in groupoidification - the path integral;

• and something similar or is indicated in section 3 and section 6 of

  • Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups (arXiv:0905.0731)

based on material (on categories of “families”) in On the Classification of Topological Field Theories .