Quantum field theory 2, lecture 25
Implications of symmetries
Besides unitarity and translational symmetries one can also make use of the Lorentz group in order to constrain the spectral function \(\Delta^\rho_{ab}(z, \mathbf{p})\). For example, when \(a\) and \(b\) are indices for scalar fields we infer immediately that \(\Delta^\rho_{ab}(z, \mathbf{p})\) must be a scalar under Lorentz transformations and can depend on \(\mathbf{p}\) trough its magnitude or \(\mathbf{p}^2\), only. When \(a\) and \(b\) correspond to spinor, vector or tensor fields it is best to decompose them into irreducibles with respect to the rotation group to find a convenient form for the spectral function.
For the vacuum state at vanishing temperature and density, \(T=\mu= 0\), we are dealing with a Lorentz invariant situation. Instead of using a spectral representation with an integral over a frequency it is then much more elegant to write it as in integral over a mass parameter. The latter is a singlet under the Lorentz group, while \(z\) is only a singlet under the part of it (rotations) that remains unbroken in the fluid rest frame.
Källén-Lehmann spectral representation in vacuum
In vacuum Lorentz symmetry implies that \(\Delta_{ab}^\rho(\omega, \mathbf{p})\) can, for \(a\) and \(b\) labeling scalar fields, only depend on \(-\omega^2+\mathbf{p}^2\) and \(\text{sign}(\omega)\). For spinor, vector or tensor fields the analysis is slightly more involved but by decomposing the fields into irreducible representations one can eventually arrive at expressions very similar to the scalar case.
Concentrating on vaccum fields, one can write the complex argument Greens function as an integral over a mass squared parameter, \[\begin{split} G_{ab}(p) = & \int_0^\infty d\mu^2 \,\rho_{ab}(\mu^2) \; \frac{1}{p^2 + \mu^2} \\ = & \int_0^\infty d\mu^2 \, \rho_{ab}(\mu^2) \left[ \frac{1}{\sqrt{\mathbf{p}^2+\mu^2}-\omega} - \frac{1}{-\sqrt{\mathbf{p}^2+\mu^2}-\omega} \right] \frac{1}{2\sqrt{\mathbf{p^2+\mu^2}}}, \end{split}\] which is known as the Källén-Lehmann representation.
Performing the variable substitution \(z=\sqrt{\mathbf{p}^2+\mu^2}\) one can rewrite this as \[G_{ab}(p) = \int_0^\infty dz \, \rho_{ab}(z^2 - \mathbf{p}^2) \left[ \frac{1}{z-\omega} - \frac{1}{-z-\omega} \right].\] We can write this as in integral over \(z\) in the range \((-\infty, \infty)\) in the form discussed previously when we identify \[\Delta^\rho_{ab}(\omega, \mathbf{p}) = 2\pi \, \text{sign}(\omega) \, \rho_{ab}(\omega^2-\mathbf{p}^2). \label{eq:vacuumLimitRho}\] As before, one can immediatley obtain the retarded, advanced, Feynman or Matsubara propagator by specializing to the approriate frequency domain.
Complex conjugate fields
So far our discussion was very general and we did not assume much about the fields for which we considered the correlation functions. For special cases one can make further going statements. In particular for two complex conjugate fields, \(\chi_a(x) = \varphi(x)\) and \(\chi_b(y) = \varphi^*(y)\), we find \[\Delta^\rho_{ab}(p) = \sum_{m,l} \delta^{(4)}(p-p_l+p_m) \frac{1}{Z} \left(e^{p_m \beta} \mp e^{p_l\beta} \right) |\langle m | \varphi(0) | l \rangle|^2,\] which clearly shows that the spectral density must be real, \[\Delta^\rho_{ab}(p) \in \mathbb{R}.\] In a similar way one can also find reality constraints for spinor, vector or tensor fields.
Sum rules
If one field is the canonical conjugate momentum field of the other, e. g. \(\chi_a(x)=\varphi(x)\) and \(\chi_b(y) = - i\Pi(y)\), they fulfill a canonical commutation relation at equal time, \[_{\mp} = i \delta^{(3)}(\mathbf{x}-\mathbf{y})\] or \[_\mp = \delta^{(3)}(\mathbf{x}-\mathbf{y}).\] This impies for the spectral function \[\Delta^\rho_{ab}(0,\mathbf{x}-\mathbf{y}) = \langle [\chi_a(t,\mathbf{x}), \chi_b(t,\mathbf{y})]_\mp \rangle = \int \frac{d\omega}{2\pi} \int_\mathbf{p} e^{i\mathbf{p}(\mathbf{x}-\mathbf{y})} \Delta^\rho_{ab}(p) = \delta^{(3)}(\mathbf{x}-\mathbf{y}).\] This only works when \[\int_{-\infty}^\infty \frac{dz}{2\pi} \Delta^\rho_{ab}(z,\mathbf{p}) = 1,\] for any value of \(\mathbf{p}\), which is called a sum rule. It is a direct consequence of the non-perturbative spectral representation and the canonical commutation relation.
Remarks
Our discussion did not rely on perturbation theory and the spectral representation is a non-perturbative statement.
We did use unitarity at the fundamental level, to have a Heisenberg representation.
We also used translation invariance in time, which is closely related to the assumption of a thermal equilibrium or vacuum state. Translation invariance in space has also been used but could have been avoided.
Assuming real spectral function one finds \[\Delta^\rho_{ab}(\omega, \mathbf{p}) = 2\, \text{Im}\left( \Delta^R_{ab}(p) \right) = - 2\, \text{Im}\left( \Delta^A_{ab}(p) \right) = 2\, \text{sign}(\omega) \, \text{Im}\left( \Delta^F_{ab}(p) \right).\]
For a free scalar field with mass \(m\) one has simply \[\Delta^\rho(p) = 2\, \text{sign}(\omega)\, \text{Im}\left( \frac{1}{p^2+m^2-i\epsilon} \right) = 2\pi \, \text{sign}(\omega) \, \delta(p^2+m^2).\] More generally, stable particles correspond to Dirac peaks in the spectral function.
For unstable resonances the peak in the spectral function is broadened. For example, in the Breit-Wigner parametrization it becomes \[\Delta^\rho(p) = 2\, \text{sign}(\omega) \text{Im} \left( \frac{1}{p^2+m^2-im \Gamma} \right) = 2 \, \text{sign}(\omega) \frac{m\Gamma}{(p^2+m^2)^2 + m^2\Gamma^2}.\] The decay width parameter \(\Gamma\) has the interpretation of an inverse life time. The Breit-Wigner parametrization is not fully realistic, however, because \(\Delta^\rho\) does not vanish for \(\omega\to 0\).
The spectral representation constrains strongly the analytic structure of two-point functions. This can be very useful for non-perturbative investigations of quantum field theory.
The spectral representation is for two-point functions, but can also involve composite operators. In this sense it constrains also particular limits of higher-order correlation functions.
Using similar concepts (unitarity, analytic continuation in momenta) one can also investigate the analytic structure higher order correlation functions. This is known as the analytic S-matrix programm.
Analytic structure of inverse propagator
The function \(G_{ab}(p)\) is obtained via analytic continuation from the second functional derivative of the Schwinger functional \(W[J]\), if the latter is a priori defined in the Euclidean domain. Similarly, its inverse \[P_{ab}(p) = G_{ab}^{-1}(p) ,\] is obtained from the analytic continuation of the second functional derivative of the effective action \(\Gamma[\Phi]\). Similar to \(G_{ab}(p)\), the function \(P_{ab}(p)\) (or more specific its eigenvalues) might have brach cuts and zero-crossings along the axis of real frequency \(\omega\) but nowhere else.
One can decompose the inverse complex-argument propagator \[P_{ab}(p) = P_{1,ab}(p) - i s_\text{I}(\omega) \, P_{2,ab}(p) , %\label{eq:DecomposeInverseComplexArgGF}\] where \(s_\text{I}(\omega) = \text{sign}(\text{Im}(\omega))\). Both functions \(P_{1,ab}(p)\) and \(P_{2,ab}(p)\) are regular when crossing the real frequency axis. However, the sign \(s_\text{I}(\omega)\) changes, which leads to a brach cut for the function \(P_{ab}(p)\). The term \(P_{2,ab}(p)\) parametrizes the strength of the branch cut. Going back to real space one has \(\omega\to i \partial_t\) and \[s_\text{I}(\omega) = \text{sign}(\text{Im}(\omega)) \to s_\text{R}(\partial_t) = \text{sign}(\text{Re}(\partial_t)).\] It might be surprising that such terms can arise in an effective action, but it is a consequence of the spectral representation and analytic continuation.
Damping terms
It is interesting to analyse the influence of a branch cut behaviour in a simple model for one real degree of freedom \(\phi(t)\). We take the effective action to be \[\begin{split} \Gamma[\phi] = & \int d t \left\{ - \frac{1}{2} \dot \phi(t)^2 + \frac{1}{2} m \phi(t)^2 + \zeta m \, \phi(t) \, s_\text{R}(\partial_t) \dot \phi(t) \right\} \\ = & \int \frac{d \omega}{2\pi} \left\{ \frac{1}{2} \phi^*(\omega) \left[ - \omega^2 + m^2 - 2 i \, s_\text{I}(\omega) \zeta m \omega \right] \phi(\omega) \right\}. \end{split}\] From the frequency representation in the second line we can read off the inverse propagator, \[P(\omega) = 1/G(\omega) = P_1(\omega) -i s_\text{I}(\omega) P_2(\omega) = - \omega^2 + m^2 - i s_\text{I}(\omega) 2 \zeta m \omega.\] For \(\zeta=0\) there are two zero crossings of the inverse propagator at \(\omega=\pm m\), corresponding to poles of the propagator. However, for \(\zeta>0\) there is instead a branch cut extending along the real frequency axis, except for the point \(\omega=0\) where \(P_2=0\).
The inverse retarded propagator is obtained by eveluating this just above the real axis, i. e. for \(s_\text{I}(\omega) = \text{sign}(\text{Im}(\omega))=1\). This gives \[1/\Delta^R(\omega) = -\omega^2 -2i \zeta m \omega + m^2.\] Interpreted as a field equation, this corresponds to \[\phi(t) = 0.\] This is the equation of motion of a damped harmonic oscillator where \(\zeta\) is known as the damping ratio! The sign of the damping term would have been opposite if we had considered the inverse advanced propagator instead. We conclude that brach cuts can be associated with dissipative behaviour.
Fluctuation-dissipation relation for damped harmonic oscillator
For the damped harmonic oscillator we find the spectral density \[\Delta^\rho(\omega) = 2\, \text{Im}\left( \frac{1}{-\omega^2+m^2 - 2i \zeta m\omega} \right) = \frac{4 \zeta m\omega}{(\omega^2-m^2)^2 + 4 \zeta^2 m^2 \omega^2}.\] Note that this vanishes indeed for \(\omega\to 0\) as it should.
From the fluctuation-dissipation relation we find the statistical correlation function \[\Delta^S(\omega) = \left[ \frac{1}{2} + n_B(\omega) \right] \frac{4 \zeta m\omega}{(\omega^2-m^2)^2 + 4 \zeta^2 m^2 \omega^2}.\] Exercise: Work out the corresponding correlation function as a function of time difference by Fourier transform and discuss their physical significance.
Linear response
Let us now discuss an experimental situation where we conider the reaction of an expectation value to a change in our field theory at an earlier time. When this reaction or response is small, as a result of a small change or perturbation, one can describe it by linear response theory, that we will develop now.
For concreteness, we assume that the expectation value we observe is \(\langle \chi_a(x)\rangle\) where \(\chi_a(x)\) is some bosonic field, that can be fundamental or composite. Typical examples are an order parameter field \(\phi(x)\), the componets of an electromagnetic current \(J^\mu(x)\), or of the energy-momentum tensor \(T^{\mu\nu}(x)\). Without loss of generality we can assume that the expectation value vanishes in thermal equilibrium, and consider only the modification as a result of the perturbation.
Concerning the perturbation, we shall assume it is given by a change in the action \[\Delta S[\phi] = \int d^d y \left\{ j_b(y) \chi_b(y) \right\},\] where \(\chi_b(y)\) are also the components of some field (fundamental or composite) and the source \(j_b(y)\) parametrises the strength of the perturbation. We are interested in a linear term, which can be of the form \[\bar \chi_a(x) = \langle \chi_a(x) \rangle = \int d^d y \left\{ \Delta^R_{ab}(x,y) j_b(y) \right\}.\] The index \(R\) is here for “response”, but we will see later that \(\Delta^R_{ab}(x,y)\) is in fact a retarded correlation function so that the notation is in agreement with notation introduced earlier.
The expectation value could be corrected by higher order terms in the source, but for a stable thermal equilibrium situations one can always make \(j_b(y)\) small enough so that the linear term fully dominates.
Translational invariance, causality, Fourier representation
If the equilibrium state i question has translational symmetries in time and in space, the response function can be a function of the coordinate differences only, \(\Delta_{ab}(x,y) = \Delta_{ab}(x-y)\). Moreover, by causality it must vanish for \(y^0>x^0\) or more generally whenever \(y\) is not in the past light cone of \(x\).
It is convenient to introduce the Fourier representation \[\Delta^R_{ab}(x-y) = \int \frac{d^d p}{(2\pi)^d} e^{ip(x-y)} \Delta^R_{ab}(p).\] The causality condition implies that \(\Delta_{ab}(p)\) must be an analytic function in the upper half of the complex frequency plane. One can write now \[\bar \chi_a(\omega, \mathbf{p}) = \Delta^R_{ab}(\omega, \mathbf{p}) j_b(\omega, \mathbf{p}),\] where \(j_b(\omega,\mathbf{p})\) and \(\bar \chi_a(\omega, \mathbf{p})\) are the source and signal in Fourier space, respectively. A periodic driving through the source induces a periodic signal of the same frequency. The generation of higher harmonics would correspond to non-linear response.
Gauge invariance and conservation laws
The source \(j_b(y)\) is sometimes a gauge field and the responding field might be a conserved current. This has implications for the response function. For example, an electromagnetic current \(J^\mu(x)\) might be induced as a response to a perturbation in the electromagnetic gauge field \(A_\nu(y)\), \[\delta J^\mu(x) = \int d^dy\left\{ \Delta^{\mu\nu}_R(x-y) \delta A_\nu(y) \right\},\] with electromagnetic retarded response function \(\Delta^{\mu\nu}_R(x-y)\). The current should be unaffected by a gauge transformation of the form \(A_\nu(y) \to A_\nu(y)+\partial_\nu \alpha(y)\), or \[0 = \int d^dy\left\{ \Delta^{\mu\nu}_R(x-y) \partial_\nu \delta\alpha(y) \right\} = \int d^dy\left\{ \delta\alpha(y) \nabla_\nu \Delta^{\mu\nu}_R(x-y) \right\}.\] Because \(\alpha(y)\) is arbitrary this implies \[\nabla_\nu \Delta^{\mu\nu}_R(x-y)=0.\] Similarly, the induced current should be conserved, \(\nabla_\mu J^\mu(x)=0\), which implies \[\nabla_\mu \Delta^{\mu\nu}_R(x-y)=0.\] In a similar way one can work out implications of gauge invariance, or Ward-Takahashi identities, for many response functions.