# Källen-Lehmann Representation

#### lonelyphysicist

I have two basic questions about the full propagator (2-point function) in QFT. Am I correct that for a scalar field, it is

$$\frac{iZ}{p^{2}-m^{2}+i \epsilon} + \int_{m^{2}}^{\infty} dX \frac{\rho[X]}{p^{2}-X+i \epsilon} ?$$

(1) Is this form of the propagator a feature of _quantum_ field theory? What if we have a nonlinear classical field theory? Would there still be something like that? Maybe Z = 1 (I'm not even sure about this) but perhaps we'd still have the term involving the integral?

(2) In QFT we seem to compute Z iteratively -- we compute 2-point function iteratively, up to a given number of loops -- and then we introduce Z's and mu's (if we're doing dimensional regularization). What about the term involving the integral? I don't recall it ever coming up except when the Källen-Lehmann rep is mentioned. Also, what about bound states; where does it come in?

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#### DarMM

Gold Member
I have two basic questions about the full propagator (2-point function) in QFT. Am I correct that for a scalar field, it is

$$\frac{iZ}{p^{2}-m^{2}+i \epsilon} + \int_{m^{2}}^{\infty} dX \frac{\rho[X]}{p^{2}-X+i \epsilon} ?$$
Yes, that is correct. $\rho\left(X\right)$, typically denoted $\rho\left(M^2\right)$, is often a sum of delta functions for isolated particles and bound states and some continuous function for the higher mass spectrum.

lonelyphysicist said:
(1) Is this form of the propagator a feature of _quantum_ field theory? What if we have a nonlinear classical field theory? Would there still be something like that? Maybe Z = 1 (I'm not even sure about this) but perhaps we'd still have the term involving the integral?
Nothing like this at all. In a classical field theory one simply has the single configuration that occurs so something like
$$\langle \phi(x)\phi(y)\rangle$$
is just
$$\phi(x)\phi(y)$$
That is the product of the field values at those two points.

lonelyphysicist said:
(2) In QFT we seem to compute Z iteratively -- we compute 2-point function iteratively, up to a given number of loops -- and then we introduce Z's and mu's (if we're doing dimensional regularization). What about the term involving the integral? I don't recall it ever coming up except when the Källen-Lehmann rep is mentioned. Also, what about bound states; where does it come in?
It will come up as soon as you move beyond tree level, whereupon the two-point function will have a more complex form than
$$\frac{i}{p^2 - m^2 +i\epsilon}$$

Bound states show up as I mentioned above, delta functions in $\rho\left(M^2\right)$, and contribute terms like
$$\frac{i}{p^2 - m_{b}^{2} +i\epsilon}$$
where $m_b$ is the bound state mass.

Typically though bound states are higher in energy than the two particle threshold which gives the continuous part of $\rho\left(M^2\right)$ and so you wouldn't typically see this term in an obvious fashion after doing perturbative computations.

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