# The Heisenberg picture and causality

## Schrödinger picture

In the Schrödinger picture of the Klein-Gordon field, the state of the system at any one time \(t\) is described by a vector \(|\psi(t)\rangle\) in a Hilbert space. As time varies, the state vector evolves according to the Schrödinger equation: \[i\frac{d}{dt}|\psi(t)\rangle=H|\psi(t)\rangle,\] where \(H\) is the quantum Hamiltonian operator and we are working in units where \(\hbar=1\). The solution of this equation with initial state \(|\psi(0)\rangle\) is \[|\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle,\] assuming the system is autonomous, that is the Hamiltonian has no time dependence.

## The Heisenberg picture

The Heisenberg picture emphasises instead the operators \(A\) as dynamical objects which vary in time according to the Heisenberg equation \[\frac{d}{dt}A(t)=i[H,A(t)].\] If \(H\) is autonomous then a solution is given by \(A(t)=e^{iHt}A(0)e^{-iHt}\). Note that if \(|\psi(t)\rangle\) solves the Schrödinger equation and \(A(t)\) solves the Heisenberg equation, then \[\langle\psi(0)|A(t)|\psi(0)\rangle=\langle\psi(t)|A(0)|\psi(t)\rangle,\] so the two pictures are equivalent as far as computing expectation values is concerned.

## Our favourite operators in the Heisenberg picture

For the Klein-Gordon system, the creation and annihilation operators, \(a_\mathbf{p}^\dagger\) and \(a_{\mathbf{p}}\), satisfy the following commutation relations with the Hamiltonian \begin{align*} [H,a_{\mathbf{p}}]&=-\omega_{\mathbf{p}}a_{\mathbf{p}},\\ [H,a^{\dagger}_{\mathbf{p}}]&=\omega_{\mathbf{p}}a^{\dagger}_{\mathbf{p}}. \end{align*} The Heisenberg equation becomes \begin{align*} \frac{d}{dt}a_{\mathbf{p}}(t)&=-i\omega_{\mathbf{p}}a_{\mathbf{p}},\\ \frac{d}{dt}a^{\dagger}_{\mathbf{p}}(t)&=-i\omega_{\mathbf{p}}a^{\dagger}_{\mathbf{p}} \end{align*} and the solutions are \begin{align*} a_{\mathbf{p}}(t)&=e^{-i\omega_{mathbf{p}}}a_{\mathbf{p}}(0),\\ a^{\dagger}_{\mathbf{p}}(t)&=e^{-i\omega_{mathbf{p}}}a^{\dagger}_{\mathbf{p}}(0). \end{align*} The operators \(\phi(x)=\phi(\mathbf{x},t)\) and \(\pi(x)=\pi(\mathbf{x},t)\) are therefore given by \begin{align*} \phi(x)&=\int\frac{d^3 p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{\mathbf{p}}}}\left(a_{\mathbf{p}}(0)e^{-ip\cdot x}+a^{\dagger}_{\mathbf{p}}(0)e^{ip\cdot x}\right)\\ \pi(x)&=-i\int\frac{d^3 p}{(2\pi)^3}\sqrt{\frac{\omega_{\mathbf{p}}}{2}}\left(a_{\mathbf{p}}(0)e^{-ip\cdot x}-a^{\dagger}_{\mathbf{p}}(0)e^{ip\cdot x}\right) \end{align*} where \(p=(\omega_{\mathbf{p}},\mathbf{p})\) and \(p\cdot x=\omega_{\mathbf{p}}t-\mathbf{p}\cdot\mathbf{x}\).

## Causality

Recall that \[ [a_{\mathbf{p}},a^{\dagger}_{\mathbf{q}}]=(2\pi)^3\delta(\mathbf{p}-\mathbf{q}).\] Using this commutation relation, we have \begin{align*} [\phi(x),\phi(y)]&= \int\frac{d^3 p}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{\mathbf{p}}}} \int\frac{d^3 q}{(2\pi)^3}\frac{1}{\sqrt{2\omega_{\mathbf{q}}}} \left([a_{\mathbf{p}},a^{\dagger}_{\mathbf{q}}]e^{iq\cdot y-p\cdot x} +[a^{\dagger}_{\mathbf{p}},a_{\mathbf{q}}]e^{ip\cdot x-q\cdot y}\right)\\ &=\int\frac{d^3 p}{(2\pi)^3}\frac{1}{2\omega_{\mathbf{p}}}\left(e^{ip\cdot(y-x)}-e^{ip\cdot(x-y)}\right)\\ &=D(x-y)-D(y-x), \end{align*} where we have defined \[D(z):=\int\frac{d^3 p}{(2\pi)^3}\frac{1}{2\omega_{\mathbf{p}}}e^{-ip\cdot z}.\] The function \(D(z)\) is Lorentz invariant (the integration measure \(d^3 p/2\omega_{\mathbf{p}}\) is the Lorentz invariant measure on the mass shell and the integrand \(e^{-ip\cdot z}\) is certainly Lorentz invariant). If \(x\) and \(y\) are spacelike separated (\((x-y)^2<0\)), then, after a Lorentz transformation, we can assume that they lie on the spacelike slice \(t=0\). In that case the integrals \(D(x-y)\) and \(D(y-x)\) are equal when we make the change of variables \(\mathbf{p}\to-\mathbf{p}\), so, if \((x-y)^2<0\) we get \[ [\phi(x),\phi(y)]=0.\] The quantum Klein-Gordon field is therefore causal: local operators at spacelike separations commute.

Note that the amplitude \[\langle 0|\phi(x)\phi(y)|0\rangle\] for the
process "particle is created at \(y\) and moves to \(x\) where it is
destroyed" is equal to \(D(x-y)\). This is actually nonzero
(exponentially decaying) outside the lightcone and oscillatory inside
the lightcone. However, when \((x-y)^2<0\), this amplitude is
cancelled by the process "particle is created at \(y\) and moves to
\(x\) where it is destroyed", as the expectation value of the
*commutator* \([\phi(x),\phi(y)]\) is zero.

## The propagator

Consider the operators \(\phi(x)\) and \(\phi(y)\).

- The time-ordered product \(T\phi(x)\phi(y)\) is either \(\phi(x)\phi(y)\) or \(\phi(y)\phi(x)\) (if \(x^0>y^0\) or \(y^0\leq x^0\) respectively).
- The normal-ordered product \(:\phi(x)\phi(y):\) is the result of expressing \(\phi(x)\) and \(\phi(y)\) in terms of creation and annihilation operators, multiplying them together, and then, in each monomial built out of creation and annihilation operators, blithely moving all the creation operators to the left.
- Let \(\Delta_F(x-y)=\begin{cases}D(y-x)&\mbox{ if
}x^0>y^0,\\D(x-y)&\mbox{ if }y^0\geq x^0\end{cases}\). This is
called the
*Feynman propagator*.