The Källén–Lehmann spectral representation, or simply Lehmann representation, gives a general expression for the (time ordered) two-point function of an interacting quantum field theory as a sum of free propagators. It was discovered by Gunnar Källén in 1952, and independently by Harry Lehmann in 1954. This can be written as, using the mostly-minus metric signature,
Δ ( p ) = ∫ 0 ∞ d μ 2 ρ ( μ 2 ) 1 p 2 − μ 2 + i ϵ , {\displaystyle \Delta (p)=\int _{0}^{\infty }d\mu ^{2}\rho (\mu ^{2}){\frac {1}{p^{2}-\mu ^{2}+i\epsilon }},}
where ρ ( μ 2 ) {\displaystyle \rho (\mu ^{2})} is the spectral density function that should be positive definite. In a gauge theory, this latter condition cannot be granted but nevertheless a spectral representation can be provided. This belongs to non-perturbative techniques of quantum field theory.
Mathematical derivation The following derivation employs the mostly-minus metric signature. In order to derive a spectral representation for the propagator of a field Φ ( x ) {\displaystyle \Phi (x)} , one considers a complete set of states { | n ⟩ } {\displaystyle \{|n\rangle \}} so that, for the two-point function one can write
⟨ 0 | Φ ( x ) Φ † ( y ) | 0 ⟩ = ∑ n ⟨ 0 | Φ ( x ) | n ⟩ ⟨ n | Φ † ( y ) | 0 ⟩ . {\displaystyle \langle 0|\Phi (x)\Phi ^{\dagger }(y)|0\rangle =\sum _{n}\langle 0|\Phi (x)|n\rangle \langle n|\Phi ^{\dagger }(y)|0\rangle .}
We can now use Poincaré invariance of the vacuum to write down
⟨ 0 | Φ ( x ) Φ † ( y ) | 0 ⟩ = ∑ n e − i p n ⋅ ( x − y ) | ⟨ 0 | Φ ( 0 ) | n ⟩ | 2 . {\displaystyle \langle 0|\Phi (x)\Phi ^{\dagger }(y)|0\rangle =\sum _{n}e^{-ip_{n}\cdot (x-y)}|\langle 0|\Phi (0)|n\rangle |^{2}.}
Next we introduce the spectral density function
ρ ( p 2 ) θ ( p 0 ) ( 2 π ) − 3 = ∑ n δ 4 ( p − p n ) | ⟨ 0 | Φ ( 0 ) | n ⟩ | 2 {\displaystyle \rho (p^{2})\theta (p_{0})(2\pi )^{-3}=\sum _{n}\delta ^{4}(p-p_{n})|\langle 0|\Phi (0)|n\rangle |^{2}} . Where we have used the fact that our two-point function, being a function of p μ {\displaystyle p_{\mu }} , can only depend on p 2 {\displaystyle p^{2}} . Besides, all the intermediate states have p 2 ≥ 0 {\displaystyle p^{2}\geq 0} and p 0 > 0 {\displaystyle p_{0}>0} . It is immediate to realize that the spectral density function is real and positive. So, one can write
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