In mathematical physics, some approaches to quantum field theory are more popular than others. For historical reasons, the Schrödinger representation is less favored than Fock space methods. In the early days of quantum field theory, maintaining symmetries such as Lorentz invariance, displaying them manifestly, and proving renormalisation were of paramount importance. The Schrödinger representation is not manifestly Lorentz invariant and its renormalisability was only shown as recently as the 1980s by Kurt Symanzik (1981).
The Schrödinger functional is, in its most basic form, the time translation generator of state wavefunctionals. In layman's terms, it defines how a system of quantum particles evolves through time and what the subsequent systems look like.
Quantum mechanics is defined over the spatial coordinates
x
\hat{x}\psi(x)=x\psi(x)
\psi(x)=\langlex|\psi\rangle
\hat{x}\left|x\right\rangle=x\left|x\right\rangle
i\partial0\left|\psi(t)\right\rangle=\hat{H}\left|\psi(t)\right\rangle
However, in quantum field theory, the coordinate is the field operator
\hat{\phi}x=\hat{\phi}(x)
\hat{\phi}(x)\Psi\left[\phi( ⋅ )\right]=\operatorname\phi\left(x\right)\Psi\left[\phi( ⋅ )\right],
\Psi\left[\phi( ⋅ )\right]=\left\langle\phi( ⋅ )|\Psi\right\rangle
is obtained by means of the field eigenstates
\hat{\phi}(x)\left|\Phi( ⋅ )\right\rangle=\Phi(x)\left|\Phi( ⋅ )\right\rangle,
\Phi( ⋅ )
i\partial0\left|\Psi(t)\right\rangle=\hat{H}\left|\Psi(t)\right\rangle.
Thus the state in quantum field theory is conceptually a functional superposition of field configurations.
\hat{\phi}(x)
\phi(x)
\hat{\phi}(x)\left|\phi\right\rangle=\phi\left(x\right)\left|\phi\right\rangle
\left|\phi\right\rangle\propto
| ||||||
| e | ||||||
| + |
(x))2
\hat{\Phi}+\left(x\right)
\hat{\phi}\left(x\right)
| \dagger | |
| a | |
| k |
For a time-independent Hamiltonian, the Schrödinger functional is defined as
l{S}[\phi2,t2;\phi1,t1]=\langle\phi
| -iH(t2-t1)/\hbar | |
| 2|e |
|\phi1\rangle.
In the Schrödinger representation, this functional generates time translations of state wave functionals, through
\Psi[\phi2,t2]=\intl{D}\phi1l{S}[\phi2,t2;\phi1,t1]\Psi[\phi1,t1].
The normalized, vacuum state, free field wave-functional is the Gaussian
\Psi0[\phi]=
| |||||
| \det{} | \left( |
| K | |
| \pi |
\right)
| |||||
| e |
\intd\vec{y}\phi(\vec{x})K(\vec{x},\vec{y})\phi(\vec{y})}=
| |||||
| \det{} | \left( |
| K | |
| \pi |
\right)
| |||||
| e |
,
where the covariance K is
K(\vec{x},\vec{y})=\int
| d3k | |
| (2\pi)3 |
\omega\vecei ⋅ (\vec{x}-\vec{y})}.
This is analogous to (the Fourier transform of) the product of each k-mode's ground state in the continuum limit, roughly (Hatfield 1992)
\Psi0[\tilde\phi]=\lim\Delta \prod\vec\left(
| \omega\vec | |
| \pi |
| ||||
| \right) |
| ||||||||
| e |
\to\left(\prod\vec\left(
| \omega\vec | |
| \pi |
| ||||
| \right) |
\right)
| ||||||||
| e |
.
Each k-mode enters as an independent quantum harmonic oscillator. One-particle states are obtained by exciting a single mode, and have the form,
\Psi[\phi]\propto\intd\vec{x}\intd\vec{y}\phi(\vec{x})K(\vec{x},\vec{y})f(\vec{y})\Psi0[\phi]=\phi ⋅ K ⋅ f
| |||||
| e |
.
For example, putting an excitation in
\vec{k}1
\Psi1[\tilde\phi]=\left(
| |||||||
| (2\pi)3 |
| ||||
| \right) |
\tilde\phi(\vec{k}1)\Psi0[\tilde\phi]
\Psi1[\phi]=\left(
| |||||||
| (2\pi)3 |
| ||||
| \right) |
\intd3
| -i\vec{k | |
| ye | |
| 1 |
⋅ \vecy}\phi(\vecy)\Psi0[\phi].
(The factor of
(2\pi)-3/2
\Deltak=1
For clarity, we consider a massless Weyl–Majorana field
\hat\psi(x)
\Psi[u]
\hat\psi(x)
\hat\psi(x)|\Psi\rangle=
| 1 | |
| \sqrt2 |
\left(u(x)+
| \delta | |
| \deltau(x) |
\right)|\Psi\rangle.