Schwinger proper time formalism

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Contents

Motivation

Let us attempt to solve the free scalar field classically, i.e., obtain its Green's functions. For the free field, these Green's functions turn out to be the correct two-point functions for the quantum field also. Thus we could also consider a free quantum field interacting with a classical background field. We will begin with a metric signature of (-,+,+,+)\,. Then, in Euclidean space, i.e., after Wick rotation, we are tasked with solving

(-\partial_x^2 + m^2) \langle 0|\phi(x)\phi(y)|0\rangle = \delta(x-y)\,,

i.e.,

(-\partial_x^2 + m^2) G(x-y) = \delta(x-y)\,.

We can replace this differential equation with a differential equation with better smoothness properties, so we appeal to the heat kernel method: We introduce a new parameter \tau\, and let

G(x) = i\int_0^\infty \!d\tau g(x, \tau)\,,

where

(-\partial^2 + m^2)g(x,y,\tau) = i \frac{\partial}{\partial \tau} g(x,y,\tau)\,,

supplemented by the initial conditions g(x,y,0) = \delta(x-y)\, and g(x,y,\infty) = 0\,. Formally this is only a mathematical procedure towards solving the classical field theory. However, this equation has the form of a Schrödinger equation with Hamiltonian H = (-\partial^2 + m^2)\,, and we wish to solve

i \frac{\partial}{\partial \tau} g(x,y,\tau) = H g(x,y,\tau)\,,

where we now interpret g(x,y,\tau)\, as a wavefunction[1] of a single particle moving through Minkowski space augmented by an extra parameter, i.e., M\otimes \mathbb{R}\,. In other words, the "position" of the particle is a point in spacetime. We have at our disposal all the mathematical methods of quantum mechanics. Thus, formally,

G(x,y) = i \left\langle x \left| \int_0^\infty\!d\tau\, e^{-i H \tau } \right| y \right\rangle\,,

or

\frac{1}{H} = i \int_0^\infty\!d\tau\, e^{-i H \tau }\,.


In particular, let the particle's state at time \tau = 0\, be \left|\psi(0)\right\rangle = \left|y\right\rangle\,, so that its state at time \tau\, is \psi(\tau) = e^{-i H \tau} \left|\psi(0)\right\rangle\,. The wavefunction at time \tau\, is therefore

g(x,y,\tau)\, = \left\langle x | \psi(\tau) \right\rangle\,
= \left\langle x \right| e^{-i H \tau} \left|y\right\rangle\,,
= \int\!\mathcal{D}p \mathcal{D}x e^{i \int\!d\tau( p \dot{x} - H)}\,,
= \int\!\mathcal{D}x e^{i \int\!d\tau L(x, \dot{x}) }\,.
= \int\!\mathcal{D}x e^{iS[x]}\,.

We can also interpret this as doing quantum field theory on the world-line of a particle where the field x^\mu(\tau)\, is defined on the 1-D manifold of the world-line. While such an interpretation is perhaps overkill here, this viewpoint is appropriate for string theory and is certainly instructive.

Point particle action

From [x^\mu, -i\partial_\mu]\psi(x) = i \psi(x)\, we deduce that p_\mu = -i \partial_\mu\, in the position representation. From Hamilton's equation \dot{x} = \frac{\partial H}{\partial p}\, we deduce that \dot{x}^\mu = 2 \delta^{\mu\nu}p_\nu\, (remembering that we are in Euclidean space). Then the Legendre transform gives the Lagrangian L = \frac{\dot{x}^\mu \dot{x}^\mu}{4} - m^2\,. The action is therefore the gauge fixed version of

S[x^\mu, e] = \frac{1}{2}\int\!d\tau\left( e^{-1} \dot{x}^2 - e m^2\right)\,

with e = 2\,. Solving the equation of motion for e\, gives e = \frac{1}{m}\sqrt{-\dot{x}^\mu \dot{x}^\mu}\,, and substituting into the action gives the action of the relativistic point particle:

S[x^\mu] = - m \int\!d\tau\, \sqrt{-\dot{x}^\mu \dot{x}^\mu}\,.

If we translate back to Minkowski space, then

S[x^\mu] = - m \int\!d\tau\, \sqrt{-\dot{x}^\mu \dot{x}_\mu}\,,

where the metric again has signature (-,+,+,+)\,. Thus, by solving the classical free scalar field, and also the quantum version (since it is non-interacting), we are lead to quantize the relativistic point particle. Notice that, expressed in this form, \tfrac{1}{m}\, plays the role of \hbar\, in quantum mechanics.

Heuristic derivation

Path integral

See also

References

[2] [3] [4] [5] [6] [7]

  1. ↑ We try not to take the analogy too far. H\, is not bounded from below.
  2. ↑ V.A. Fock (1937). "{{{title}}}". Izvestiya Akad. Nauk USSR, OMEN: 557. 
  3. ↑ V. A. Fock (1937). "Die Eigenzeit in der klassischen und in der Quantenmechanik". Phyz. Z. Sow. 12: 404–425. 
  4. ↑ Julian Schwinger (1951). "On Gauge Invariance and Vacuum Polarization". Phys. Rev. 82: 664-679. 
  5. ↑ R.P. Feynman (1950). "Mathematical Formulation of the Quantum Theory of Electromagnetic Interaction". Phys. Rev. 80: 440. DOI:10.1103/PhysRev.80.440. 
  6. ↑ R.P. Feynman (1951). "An Operator Calculus Having Applications in Quantum Electrodynamics". Phys. Rev. 84: 108. DOI:10.1103/PhysRev.84.108. 
  7. ↑ B. S. DeWitt (1965). Dynamical Theory of Groups and Fields. Gordon and Breach. 
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