Källén-Lehmann spectral representation

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Consider the two-point function $\left\langle 0 | T \phi(x) \phi(y) | 0 \right\rangle\,$ of the interacting scalar field. We wish to insert the completeness relation between the two fields in this expression, where the states are taken to be eigenstates of the momentum operator $\mathbf{P}\,$ with eigenvalues $\mathbf{k}\,$ . We will assume that the theory is Lorentz invariant so that $\left[ H, \mathbf{P} \right] = 0\,$ and states can be chosen to be simultaneous energy eigenstates also. We will also assume that there is a unique vacuum state $\left|0\right\rangle\,$ with $P^\mu \left|0\right\rangle = 0\,$ , from which it also follows that $J^{\mu\nu} \left|0\right\rangle = 0\,$ . Then, let

$1 = \left | 0 \right\rangle \left\langle 0 \right| + \sum_n \int\!\frac{d^3k}{(2\pi)^3} \frac{1}{2 E_\mathbf{k}(n)} \left | n_\mathbf{k} \right\rangle \left\langle n_\mathbf{k}\right|\,$ ,

where we have grouped all the states in the Hilbert space according to their momenta $\mathbf{k}\,$ , and we have normalized the states in a Lorentz invariant way. In other words, $E_\mathbf{k}(n) = \sqrt{\mathbf{k}^2 + m_n^2}\,$ where $m_n\,$ is the mass of the state $\left|n_\mathbf{k}\right\rangle\,$ , or the energy of the corresponding zero-momentum state $\left|n_\mathbf{0}\right\rangle\,$ .

Assume $x^0 > y^0\,$ . Then

$\left\langle 0 | \phi(x) \phi(y) | 0 \right\rangle = \left\langle 0 | \phi(x) \left | 0 \right\rangle \left\langle 0 \right| \phi(y) | 0 \right\rangle + \sum_n \int\!\frac{d^3k}{(2\pi)^3} \frac{1}{2 E_\mathbf{k}(n)} \left\langle 0 | \phi(x) | n_\mathbf{k} \right\rangle \left\langle n_\mathbf{k}\phi(y) | 0 \right\rangle\,$ .

The first term usually vanishes by $\phi(x) \to -\phi(x)\,$ symmetry, or by Lorentz invariance ^[1], so ignoring it for now,

$\left\langle 0 | \phi(x) \phi(y) | 0 \right\rangle = \sum_n \int\!\frac{d^3k}{(2\pi)^3} \frac{1}{2 E_\mathbf{k}(n)} \left\langle 0 | \phi(x) | n_\mathbf{k} \right\rangle \left\langle n_\mathbf{k}\phi(y) | 0 \right\rangle\,$ .

Then

$\left\langle 0 \| \phi(x) \| n_\mathbf{k} \right\rangle\,$	$= \left\langle 0 \| e^{iP\cdot x}\phi(0)e^{-iP\cdot x} \| n_\mathbf{k} \right\rangle\,$ ,
	$= e^{-ik\cdot x} \left\langle 0 \| \phi(0) \| n_\mathbf{k} \right\rangle\,$ ,

where $k^0 = E_\mathbf{k}\,$ . Since the states $\left| n_\mathbf{k} \right\rangle\,$ form a complete representation of the Lorentz group, we can write $\left| n_\mathbf{k} \right\rangle = U^{-1}(\mathbf{k}) \left| n_\mathbf{0} \right\rangle\,$ where $U(\mathbf{k})\,$ is some unitary operator that implements the Lorentz boost from $\mathbf{k}\,$ to $0\,$ , so

$\left\langle 0 \| \phi(x) \| n_\mathbf{k} \right\rangle\,$	$= e^{-ik\cdot x} \left\langle 0 \| U^{-1} U \phi(0) U^{-1} U \| n_\mathbf{k} \right\rangle\,$ ,
	$= e^{-ik\cdot x} \left\langle 0 \| U^{-1} U \phi(0) U^{-1} \| n_\mathbf{0} \right\rangle\,$ ,

but $\phi(x)\,$ is a Lorentz scalar, so that $U \phi(0) U^{-1} = \phi(0)\,$ , and

$\left\langle 0 | \phi(x) | n_\mathbf{k} \right\rangle = e^{-ik\cdot x} \left\langle 0 | \phi(0) | n_\mathbf{0} \right\rangle\,$ .

Then

$\left\langle 0 | \phi(x) \phi(y) | 0 \right\rangle = \sum_n \int\!\frac{d^3k}{(2\pi)^3} \frac{1}{2 E_\mathbf{k}(n)} e^{-ik\cdot (x-y)} \left| \left\langle 0 | \phi(0) | n_\mathbf{0} \right\rangle \right|^2\,$ ,

while we can implement the time ordering by

$\left\langle 0 \| T \phi(x) \phi(y) \| 0 \right\rangle\,$	$=\theta(x^0-y^0) \left\langle 0 \| \phi(x) \phi(y) \| 0 \right\rangle + \theta(y^0-x^0) \left\langle 0 \| \phi(y) \phi(x) \| 0 \right\rangle\,$ ,
	$=\sum_n \left[ \theta(x^0-y^0) \int\!\frac{d^3k}{(2\pi)^3} \frac{1}{2 E_\mathbf{k}(n)} e^{-ik\cdot (x-y)} + \theta(y^0-x^0) \int\!\frac{d^3k}{(2\pi)^3} \frac{1}{2 E_\mathbf{k}(n)} e^{ik\cdot (x-y)} \right] \left\| \left\langle 0 \| \phi(0) \| n_\mathbf{0} \right\rangle \right\|^2\,$
	$= \sum_n \Delta_F(x-y; m_n^2) \left\| \left\langle 0 \| \phi(0) \| n_\mathbf{0} \right\rangle \right\|^2\,$ ,

where $\Delta_F(x-y; m_n^2)\,$ is the Feynman propagator for a free scalar field of mass $m_n\,$ :

$\Delta_F(x; m_n^2) = \int\!\!\frac{d^4k}{(2\pi)^4} \frac{i}{k^2 - m_n^2 + i \epsilon} e^{-ik\cdot x}$ .

Spectral density function

Introducing the spectral density function

$\rho(M^2)= \sum_n (2\pi)\delta(M^2 - m_n^2)\left| \left\langle 0 | \phi(0) | n_\mathbf{0} \right\rangle \right|^2\,$ ,

we can write the momentum space propagator of the interacting field as

$G(p) = \int_0^\infty \frac{dM^2}{2\pi} \rho(M^2) \frac{i}{p^2 - M^2 + i \epsilon}\,$ .

In other words, the propagator is a superposition of free particle propagators with different masses $M\,$ . The relative weight of each contribution is determined by the spectral density function $\rho(M^2)\,$ , which yields information of the various states $\left|n_0\right\rangle\,$ and their masses $m_n\,$ .

Analytic properties

In theories with a mass gap, one-particle states contribute an isolated delta function at $p^2 = m^2\,$ to $\rho(M^2)\,$ , while a continuum of multi-particle states begin roughly at $p^2 = 4m^2\,$ . In other words,

$\rho(M^2) = (2\pi) Z \, \delta(M^2 - m^2) + ...\,$ .

Note that $m\,$ need not equal the mass in the Lagrangian (the so-called bare mass), and is simply equal to the mass of the one-particle state. The function $G(p)\,$ has a pole in the complex $p^2\,$ -plane at $p^2 = m^2\,$ . We may also show that it has a branch cut starting roughly at $p^2 = 4m^2\,$ , i.e., where $\rho(M^2)\,$ begins to have support. Consider the integral

$\int_{\sim 4m^2}^\infty \frac{dM^2}{2\pi} \rho(M^2) \frac{i}{p^2 - M^2}\,$ ,

where $p^2 \gtrsim 4m^2\,$ . Then

$\int_{\sim 4m^2}^\infty \frac{dM^2}{2\pi} \rho(M^2) \frac{i}{p^2 - M^2 \pm i \epsilon}\,$	$=\frac{1}{2\pi i} \int_{\sim 4m^2}^\infty dM^2 \rho(M^2) \frac{1}{M^2 - p^2 \mp i \epsilon}\,$ ,
	$=\frac{1}{2\pi i} \int_{\sim 4m^2}^\infty d\mu \frac{\rho(\mu) }{\mu - p^2 \mp i \epsilon}\,$ ,
	$=\frac{1}{2\pi i} \int_{\sim 4m^2 - p^2}^\infty d\nu \frac{ \rho(\nu+p^2) }{\nu \mp i \epsilon}\,$ ,
	$=\frac{1}{2\pi i} \int_{\sim 4m^2 - p^2}^\infty d\nu \frac{ \rho(\nu+p^2) }{\nu \mp i \epsilon}\,$ ,
	$=\frac{1}{2\pi i} \left[ \mathcal{P} \int \frac{\rho(\nu + p^2) }{ \nu } \pm i \pi \rho(p^2) \right]\,$ , by the Sokhotsky-Weierstrass theorem.

The function $G(p^2)\,$ is therefore discontinuous across the real axis when $\operatorname{Re}\,p^2 \gtrsim 4m^2\,$ , indicating the presence of a branch cut.

back to unitarily inequivalent representations

on to multi-particle states

References

^[2] ^[3] ^[1]

↑ ^1.0 ^1.1 M.E. Peskin and D.V. Schroeder, An Introduction to Quantum Field Theory, Addison-Wesley, Reading, Massachusetts, 1995
↑ G. Källén (1952). "?". Helv. Phys. Acta 25: 417.
↑ H. Lehmann (1954). "Über Eigenschaften von Ausbreitungsfunktionen und Renormierungskonstanten quantisierter Felder". Nuovo Cimento 11: 342-357.

Retrieved from "http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/K%C3%A4ll%C3%A9n-Lehmann_spectral_representation"

$\left\langle 0 \| T \phi(x) \phi(y) \| 0 \right\rangle\,$	$=\theta(x^0-y^0) \left\langle 0 \| \phi(x) \phi(y) \| 0 \right\rangle + \theta(y^0-x^0) \left\langle 0 \| \phi(y) \phi(x) \| 0 \right\rangle\,$ ,
	$=\sum_n \left[ \theta(x^0-y^0) \int\!\frac{d^3k}{(2\pi)^3} \frac{1}{2 E_\mathbf{k}(n)} e^{-ik\cdot (x-y)} + \theta(y^0-x^0) \int\!\frac{d^3k}{(2\pi)^3} \frac{1}{2 E_\mathbf{k}(n)} e^{ik\cdot (x-y)} \right] \left\| \left\langle 0 \| \phi(0) \| n_\mathbf{0} \right\rangle \right\|^2\,$
	$= \sum_n \Delta_F(x-y; m_n^2) \left\| \left\langle 0 \| \phi(0) \| n_\mathbf{0} \right\rangle \right\|^2\,$ ,

Källén-Lehmann spectral representation

From Physics wiki

Spectral density function

Analytic properties

References

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