Srednicki — Quantum Field Theory · Problem 9.5

Canonical Quantization of Scalar Fields — Mark Srednicki

Physics — QFT · interaction-picture-derivation

Problem

The interaction picture. In this problem, we will derive a formula for

\langle 0|\mathrm{T}\varphi(x_n)\ldots\varphi(x_1)|0\rangle

without using path integrals. Suppose we have a hamiltonian density

\mathcal{H} = \mathcal{H}_0 + \mathcal{H}_1

, where

\mathcal{H}_0 = \frac{1}{2}\Pi^2 + \frac{1}{2}(\nabla\varphi)^2 + \frac{1}{2}m^2\varphi^2

, and

\mathcal{H}_1

is a function of

\Pi(\mathbf{x},0)

and

\varphi(\mathbf{x},0)

and their spatial derivatives. (It should be chosen to preserve Lorentz invariance, but we will not be concerned with this issue.) We add a constant to

H

so that

H|0\rangle = 0

. Let

|\emptyset\rangle

be the ground state of

H_0

, with a constant added to

H_0

so that

H_0|\emptyset\rangle = 0

. (

H_1

is then defined as

H - H_0

.) The Heisenberg-picture field is

\varphi(\mathbf{x},t) \equiv e^{iHt}\varphi(\mathbf{x},0)e^{-iHt} . \tag{9.33}

We now define the interaction-picture field

\varphi_I(\mathbf{x},t) \equiv e^{iH_0 t}\varphi(\mathbf{x},0)e^{-iH_0 t} . \tag{9.34}

a) Show that

\varphi_I(x)

obeys the Klein-Gordon equation, and hence is a free field. b) Show that

\varphi(x) = U^\dagger(t)\varphi_I(x)U(t)

, where

U(t) \equiv e^{iH_0 t}e^{-iHt}

is unitary. c) Show that

U(t)

obeys the differential equation

i\frac{d}{dt}U(t) = H_I(t)U(t)

, where

H_I(t) = e^{iH_0 t}H_1 e^{-iH_0 t}

is the interaction hamiltonian in the interaction picture, and the boundary condition

U(0) = 1

. d) If

\mathcal{H}_1

is specified by a particular function of the Schr\"odinger-picture fields

\Pi(\mathbf{x},0)

and

\varphi(\mathbf{x},0)

, show that

H_I(t)

is given by the same function of the interaction-picture fields

\Pi_I(\mathbf{x},t)

and

\varphi_I(\mathbf{x},t)

. e) Show that, for

t > 0

U(t) = \mathrm{T}\exp\!\left[-i\int_0^t dt'\, H_I(t')\right] \tag{9.35}

obeys the differential equation and boundary condition of part (c). What is the comparable expression for

t < 0

? Hint: you may need to define a new ordering symbol. f) Define

U(t_2, t_1) \equiv U(t_2)U^\dagger(t_1)

. Show that, for

t_2 > t_1

U(t_2, t_1) = \mathrm{T}\exp\!\left[-i\int_{t_1}^{t_2} dt'\, H_I(t')\right] . \tag{9.36}

What is the comparable expression for

t_1 > t_2

? g) For any time ordering, show that

U(t_3, t_1) = U(t_3, t_2)U(t_2, t_1)

and that

U^\dagger(t_1, t_2) = U(t_2, t_1)

. h) Show that

\varphi(x_n)\ldots\varphi(x_1) = U^\dagger(t_n,0)\varphi_I(x_n)U(t_n,t_{n-1})\varphi_I(x_{n-1})\ldots U(t_2,t_1)\varphi_I(x_1)U(t_1,0) . \tag{9.37}

i) Show that

U^\dagger(t_n, 0) = U^\dagger(\infty, 0)U(\infty, t_n)

and also that

U(t_1, 0) = U(t_1, -\infty)U(-\infty, 0)

. j) Replace

H_0

with

(1-i\epsilon)H_0

, and show that

\langle 0|U^\dagger(\infty,0) = \langle 0|\emptyset\rangle\langle\emptyset|

and that

U(-\infty,0)|0\rangle = |\emptyset\rangle\langle\emptyset|0\rangle

. k) Show that

\begin{aligned} \langle 0|\varphi(x_n)\ldots\varphi(x_1)|0\rangle &= \langle\emptyset|U(\infty,t_n)\varphi_I(x_n)U(t_n,t_{n-1})\varphi_I(x_{n-1})\ldots \nonumber\\ &\quad U(t_2,t_1)\varphi_I(x_1)U(t_1,-\infty)|\emptyset\rangle \times |\langle\emptyset|0\rangle|^2 . \tag{9.38} \end{aligned}

l) Show that

\langle 0|\mathrm{T}\varphi(x_n)\ldots\varphi(x_1)|0\rangle = \langle\emptyset|\mathrm{T}\varphi_I(x_n)\ldots\varphi_I(x_1)e^{-i\int d^4x\, \mathcal{H}_I(x)}|\emptyset\rangle \times |\langle\emptyset|0\rangle|^2 . \tag{9.39}

m) Show that

|\langle\emptyset|0\rangle|^2 = 1/\langle\emptyset|\mathrm{T}e^{-i\int d^4x\, \mathcal{H}_I(x)}|\emptyset\rangle . \tag{9.40}

Thus we have

\langle 0|\mathrm{T}\varphi(x_n)\ldots\varphi(x_1)|0\rangle = \frac{\langle\emptyset|\mathrm{T}\varphi_I(x_n)\ldots\varphi_I(x_1)e^{-i\int d^4x\, \mathcal{H}_I(x)}|\emptyset\rangle}{\langle\emptyset|\mathrm{T}e^{-i\int d^4x\, \mathcal{H}_I(x)}|\emptyset\rangle} . \tag{9.41}

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