Källen-Lehmann Representation

In summary, the full propagator (2-point function) in QFT for a scalar field is given by ##\frac{iZ}{p^{2}-m^{2}+i \epsilon} + \int_{m^{2}}^{\infty} dX \frac{\rho[X]}{p^{2}-X+i \epsilon}##, where ##\rho\left(X\right)## is often a sum of delta functions and a continuous function for the higher mass spectrum. This form is specific to quantum field theory and would not appear in a nonlinear classical field theory. The term involving the integral is only significant beyond tree level computations, and bound states contribute to the term through delta functions in ##\rho\
  • #1
lonelyphysicist
32
0
I have two basic questions about the full propagator (2-point function) in QFT. Am I correct that for a scalar field, it is

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

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

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

What is the Källen-Lehmann Representation?

The Källen-Lehmann Representation is a mathematical framework used in theoretical particle physics to describe the behavior of elementary particles. It is based on the concept of analyticity and provides a way to relate the physical properties of a particle, such as its mass and spin, to its interactions with other particles.

Who developed the Källen-Lehmann Representation?

The Källen-Lehmann Representation was developed independently by two physicists, Gunnar Källen and Oskar Lehmann, in the 1950s. Their work was based on previous research by Wolfgang Pauli and Pierre Weisskopf.

What is the significance of the Källen-Lehmann Representation?

The Källen-Lehmann Representation is a fundamental tool in theoretical particle physics and has been used to make predictions and calculations in various areas of the field, such as quantum field theory and particle interactions. It also provides a way to test the validity of different theoretical models.

How does the Källen-Lehmann Representation work?

The Källen-Lehmann Representation is based on the analyticity of the particle propagator, which describes the probability of a particle to propagate between two points in space and time. It involves performing integrals over a range of energies and momenta to obtain a mathematical expression that relates the particle's physical properties to its interactions.

What are some applications of the Källen-Lehmann Representation?

The Källen-Lehmann Representation has been used in various areas of theoretical particle physics, such as calculating decay rates and scattering cross sections. It has also been applied in other fields, such as condensed matter physics, to study phase transitions and critical phenomena.

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