Källén-Lehmann representation

The Källén-Lehmann representation , after Gunnar Källén and Harry Lehmann , or spectral representation is a representation of propagators in quantum field theory . The Källén-Lehmann representation is exact, i.e. it is not based on perturbation-theoretical approximations. For a scalar field in space it reads :

${\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle =\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}e^{-\mathrm {i} p(x-y)}\mathrm {i} \int _{0}^{\infty }\mathrm {d} q^{2}{\frac {\rho (q^{2})}{p^{2}-q^{2}+\mathrm {i} \epsilon }}}$

It is

• ${\displaystyle \langle \Omega |T\{\phi (x)\phi (y)\}|\Omega \rangle }$the propagator of the quantum field from space-time to and ${\displaystyle \phi }$ ${\displaystyle x}$ ${\displaystyle y}$
• ${\displaystyle \rho (q^{2})}$ the spectral density.

One defines

${\displaystyle \Pi (p^{2})=\int _{0}^{\infty }\mathrm {d} q^{2}{\frac {\rho (q^{2})}{p^{2}-q^{2}+\mathrm {i} \epsilon }}}$

as (except for numerical factors) the Fourier transform of the propagator as the propagator in momentum space.

Spectral density

The spectral density of a field is defined by ${\displaystyle \rho (p^{2})}$

${\displaystyle 2\pi \Theta (p^{0})\rho (p^{2})=\sum _{X}\prod _{j\in X}\int {\frac {\mathrm {d} ^{3}{\vec {p}}_{j}}{(2\pi )^{3}}}{\frac {1}{2p_{j}^{0}}}\delta ^{(4)}(p-p_{X})\left|\langle \Omega |\phi (0)|X\rangle \right|^{2}}$

With

• the Heaviside function , ${\displaystyle \Theta }$
• physical many-body states and ${\displaystyle |X\rangle }$
• Four-momentum of the single particles in the many-particle state. ${\displaystyle p_{j}}$

The spectral density is a Lorentz scalar and can therefore only depend on. Since only physical states are included, the spectral density follows for the physically meaningful domain . For it can be set identically to zero. Since the quantity on the right-hand side of the equation is always greater than or equal to zero, the spectral density for is also nonnegative. ${\displaystyle p^{2}}$ ${\displaystyle |X\rangle }$ ${\displaystyle p_{X}^{2}\geq 0,p_{X}^{0}>0}$ ${\displaystyle p^{2}\geq 0,p^{0}>0}$ ${\displaystyle p^{2}<0}$ ${\displaystyle p^{2}\geq 0}$

The spectral density follows from the optical theorem to

${\displaystyle \rho (p^{2})=-{\frac {1}{\pi }}\operatorname {Im} \Pi (p^{2})}$,

is therefore proportional to the imaginary part of the propagator. Since the propagator only has an imaginary part if and only if the field is on the mass shell or if the particle is heavy enough to decay into lighter particles with masses , the spectral density has a singularity at and a continuous component for . ${\displaystyle m_{1},m_{2}}$ ${\displaystyle p^{2}=m^{2}}$ ${\displaystyle p^{2}>(m_{1}+m_{2})^{2}}$

Derivation

The two-point function without time order can be activated by inserting a one in the form of ${\displaystyle \langle \Omega |\phi (x)\phi (y)|\Omega \rangle }$

${\displaystyle 1=\sum _{X}\prod _{j\in X}\int {\frac {\mathrm {d} ^{3}{\vec {p}}_{j}}{(2\pi )^{3}}}{\frac {1}{2p_{j}^{0}}}|X\rangle \langle X|}$

to be written. Further insertion of ones in the form of or the analogue with the momentum operator leads to: ${\displaystyle \exp(\mathrm {i} Px)\exp(-\mathrm {i} Px)}$ ${\displaystyle y}$ ${\displaystyle P}$

${\displaystyle \langle \Omega |\phi (x)\phi (y)|\Omega \rangle =\sum _{X}\prod _{j\in X}\int {\frac {\mathrm {d} ^{3}{\vec {p}}_{j}}{(2\pi )^{3}}}{\frac {1}{2p_{j}^{0}}}\langle \Omega |e^{\mathrm {i} Px}e^{-\mathrm {i} Px}\phi (x)e^{\mathrm {i} Px}e^{-\mathrm {i} Px}|X\rangle \langle X|e^{\mathrm {i} Py}e^{-\mathrm {i} Py}\phi (y)e^{\mathrm {i} Py}e^{-\mathrm {i} Py}|\Omega \rangle }$

From the effect of the momentum operator to the various objects - the vacuum is invariant , the momentum operator to a state gives its momentum and as a generator of translations it shifts fields - as well as the insertion of a delta function follows ${\displaystyle \exp(\mathrm {i} Px)|\Omega \rangle =|\Omega \rangle }$ ${\displaystyle \exp(\mathrm {i} Px)|X\rangle =\exp(\mathrm {i} p_{X}x)|X\rangle }$ ${\displaystyle \exp(-\mathrm {i} Px)\phi (x)\exp(\mathrm {i} Px)=\phi (0)}$

${\displaystyle \langle \Omega |\phi (x)\phi (y)|\Omega \rangle =\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}e^{-\mathrm {i} p(x-y)}\left[\sum _{X}\prod _{j\in X}\int {\frac {\mathrm {d} ^{3}{\vec {p}}_{j}}{(2\pi )^{3}}}{\frac {1}{2p_{j}^{0}}}\delta ^{(4)}(p-p_{X})\langle \Omega |\phi (0)|X\rangle \langle X|\phi (0)|\Omega \rangle \right]=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{3}}}e^{-\mathrm {i} p(x-y)}\rho (p^{2})}$

Inserting a delta distribution again results in:

${\displaystyle \langle \Omega |\phi (x)\phi (y)|\Omega \rangle =\int _{0}^{\infty }\mathrm {d} q^{2}\rho (q^{2})\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{3}}}e^{-\mathrm {i} p(x-y)}\theta (p^{0})\delta (p^{2}-q^{2})\equiv \int _{0}^{\infty }\mathrm {d} q^{2}\rho (q^{2})D(x,y,q^{2})}$

The application of the time order operator to the two-point function leads in connection with the mathematical identity

${\displaystyle D(x,y,q^{2})\theta (x^{0}-y^{0})+D(y,x,q^{2})\theta (y^{0}-x^{0})=\int {\frac {\mathrm {d} ^{4}p}{(2\pi )^{4}}}e^{\mathrm {i} p(x-y)}{\frac {\mathrm {i} }{p^{2}-q^{2}+\mathrm {i} \epsilon }}}$

for the Källén-Lehmann representation.

literature

• Matthew D. Schwartz: Quantum Field Theory and the Standard Model . Cambridge University Press, Cambridge 2014, ISBN 978-1-107-03473-0 , pp. 467-471 (English).

Individual evidence

1. ^ Gunnar Källén: On the Definition of the Renormalization Constants in Quantum Electrodynamics . In: Helvetica Physica Acta . tape 25 , no. 4 , 1952, pp. 417-434 .
2. ↑ Harry Lehmann: About properties of propagation functions and renormalization constants of quantized fields . In: Il Nuovo Cimento . tape 11 , no. 4 , 1954, pp. 342-357 .