Schrödinger functional - Alchetron, The Free Social Encyclopedia

In mathematical physics, some approaches to quantum field theory are more popular than others. For historical reasons, the Schrödinger representation is less favoured 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).

Within the Schrödinger representation, the Schrödinger wavefunctional stands out as perhaps the most useful and versatile functional tool, though interest in it is specialized at present.

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.

Example: Scalar Field

In the quantum field theory of (as example) a quantum scalar field ϕ ^ ( x ) , in complete analogy with the one-particle quantum harmonic oscillator, the eigenstate of this quantum field with the "classical field" ϕ ^ ( x ) (c-number) as its eigenvalue,

ϕ ^ ( x ) | ϕ ⟩ = ϕ ( x ) | ϕ ⟩

is (Scwartz, 2013)

| ϕ ⟩ ∝ e − ∫ d x 1 2 ( ϕ ( x ) − Φ ^ + ( x ) ) 2 | 0 ⟩

where Φ ^ + ( x ) is the part of ϕ ^ ( x ) that only includes creation operators a k † . For the oscillator, this corresponds to the representation change/map to the |x⟩ state from Fock states.

For a time-independent Hamiltonian H, the Schrödinger functional is defined as

S [ ϕ 2 , t 2 ; ϕ 1 , t 1 ] = ⟨ ϕ 2 | e − i H ( t 2 − t 1 ) / ℏ | ϕ 1 ⟩ .

In the Schrödinger representation, this functional generates time translations of state wave functionals, through

Ψ [ ϕ 2 , t 2 ] = ∫ D ϕ 1 S [ ϕ 2 , t 2 ; ϕ 1 , t 1 ] Ψ [ ϕ 1 , t 1 ] .

References

Schrödinger functional Wikipedia

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