Wick theorem in "QFT for the Gifted Amateur"

[Demystifier]

Normally I understand Wick theorem as used in particle physics, but I guess I have a problem with using it in condensed matter physics. Or at least, I have a problem with a use of Wick theorem in the book
T. Lancaster, S.J. Blundell, Quantum Field Theory for the Gifted Amateur
at page 381, Eq. (43.4)

This equation is of the form

$a^{\dagger}a^{\dagger}aa=N[a^{\dagger}a^{\dagger}aa]$
+ terms with contractions,

where $N[...]$ denotes normal ordering. However, the expression on the left $a^{\dagger}a^{\dagger}aa$ is already normal ordered, so there should be no terms with contractions on the right. The equation would make sense if different ordering (e.g. time ordering) was understood on the left, but I don't see where such different ordering comes from, given that $a^{\dagger}a^{\dagger}aa$ is essentially a Hamiltonian in Eq. (43.2).

What am I missing?

 

[Physics Monkey]

I don't seem to have the book and can't find the relevant part to preview online. I did find some other uses of normal ordering in the book which seem consistent with my usual understanding and the definition you gave.

I am also not aware of any special normal ordering prescription in condensed matter physics that would be relevant here.

My best guess is typo (or perhaps a case where the general formula trivializes) without being able to see more of the context. Sorry I can't be of more help.

Edit: the only thing I can think of is a Bogoliubov transformation where the creation and annihilation operators mix and in which normal ordering with respect to the new vacuum could look unusual in the old creation and annihilation basis.

 

[Demystifier]

Here I attach the relevant 2 pages of the book (I hope it's not against the forum rules):

 

[Demystifier]

Ah, I think I get it now.

Consider $\Phi^4$ interaction in particle physics. The interaction Lagrangian can be taken to be either $\lambda\Phi^4$ or $\lambda:\Phi^4:$. In particle physics one usually takes the second one with normal ordering, but it is also allowed to take the first one without the normal ordering. What's the difference? If one takes the interaction without the normal ordering, one gets additional Feynman diagrams, all of which have the form of bubbles without any external legs. Such bubbles (i) change the overall phase of the scattering amplitude (see e.g. Bjorken Drell) which is physically irrelevant, and (ii) change the energy of the vacuum. In general (ii) may be relevant, but in particle physics it is ignored since the vacuum energy is interpreted as the cosmological constant which is experimentally too small to be physically relevant. Therefore, in particle physics the Hamiltonian can be taken to be the one with normal ordering.

But condensed matter is different. There, the "vacuum" is interpreted as the ground state of all the electrons with energy smaller than Fermi energy. This ground-state energy has measurable physical effect, so it cannot be ignored. Therefore, in condensed matter one must take into account the contributions of all these bubbles. Indeed, at page 382 (not attached in the post above) one can see these bubble diagrams explicitly.

This is physics, but how can Eq. (43.4) be correct mathematically? Actually it is not correct as written, but only because the authors used a confusing notation. The left-hand side is really taken with some different ordering, not normal ordering. In fact, the right-hand side can be understood as a definition of the correct ordering not written explicitly on the left-hand side. Indeed, this is more clear from Eq. (43.3), which does not have a $=$ sign, but a $\rightarrow$ sign.

 

[fzero]

I think there's a way to rescue the analysis in the book. Let us recall why we use normal ordering in the first place. Normal ordering is usually introduced in the context of computing expectation values in the vacuum state. There it makes sense to define normal ordering as putting creation operators to the left of annihilation operators since then all of the terms involving normal ordered operators will vanish. Expectation values can then be computed from the rules for contracting operators (e.g. with free field theory propagators).

Now, as you say, we are in a condensed matter context where the fermion occupation numbers are partially filled. So the vacuum normal ordering doesn't make sense for expectation values in the ground state (it doesn't gain us anything since $\langle N[\cdots]\rangle_0 \neq 0$). But we can use a different definition of normal ordering. Suppose we fix the Fermi energy $E_F$. Then a useful definition of normal ordering is the following. Given the appearance of $a_k$ or $a^\dagger_k$ in an operator, if $E_k\leq E_F$, we put the creation operator on the right; if $E_k > E_F$, we put the annihilation operator on the right. This would guarantee that expectation values of normal ordered quantities would vanish analogously to the vacuum case.

I'm not sure that this is what the authors had in mind, since given the premise of the book, I'd have expected it to be spelled out in detail. One could also complain that the method is not robust, since it depends specifically on $E_F$. I agree and would prefer to simply compute expectation values by imposing delta-restrictions on the sums to determine the contractions that give appropriate number operators.

 

[vanhees71]

There's no big formal difference between condensed-matter and elementary-particle QFT. After all, both use QFT.

There are of course some differences from the fact that most condensed-matter applications are sufficiently accurately described with non-relativstic quantum field theory, which simplifies some things, because the time-oredered propagator of the Schrödinger field is also retarded, and some diagrams thus simply do not occur. Also the bound-state problem is much simpler, as the example of the hydrogen atom shows, for which you can solve the bound-state problem exactly, and that's done in any quantum-mechanics 1 lecture. The reason is that for the Schrödinger field the quantum field the mode decomposition in terms of energy-eigenstates contains only annilation operators, while in relativstic QFT you necessarily you must have both in order to build local realizations of the Poincare group, which then helps to build a Poincare invariant S-matrix etc.

On the other hand, condensed-matter theory usually uses QFT as a true many-body theory, i.e., you look at systems which contain many particles and not like in relativistic vacuum QFT as used in high-energy particle physics, with one or two particles in the initial state and a few particles in the final state, where you calculate cross sections and the like. This is done by calculating vacuum-expectation values (which task you further organize in terms of perturbation theory and put it into the elegant Feynman-diagram notation).

In many-body situations you usually work with a statistical operator to calculate expectation values. The most common applications deal with equilibrium quantum-field theory, where the statistical operator takes a quite simple form like
\begin{equation}
\label{1}
\hat{\rho}=\frac{1}{Z} \exp[-\beta (\hat{H}- \mu \hat{Q})].
\end{equation}
The partion sum $Z$ is used to properly normalize the operator,
\begin{equation}
Z=\mathrm{Tr} \; \exp[-\beta (\hat{H}- \mu \hat{Q})].
\end{equation}
Here $\hat{Q}$ means (one or more) conserved-charge operator(s) and $\mu$ the associated chemical potential(s).

Now it's clear that you can come very far by just reading the exponential operator as a time-evolution operator along an imaginary-time axis running from $0$ to $-\mathrm{i}$ in a (mathematical) complex-time plane. As a time, of course only real time makes physical sense, but the vertical line helps you to formulate perturbation theory for interacting particles in thermal equilibrium, described by the grand-canonical statistical operator (\ref{1}). You get Feynman rules pretty much the same as for vacuum theory, but with some extra rules. E.g., the tadpole diagrams you mention become relevant. In relativistic $\phi^4$ theory, e.g., there's a one-loop self-energy diagram, which (after subtracting the divergent vacuum piece) contributes to the effective mass (dependent on temperature and chemical potential(s)) of the particle in the medium. The particle becomes a "quasi-particle", which changes its mass in the medium. At higher orders, in addition to the mass the self-energy becomes complex and momentum-dependent, and the quasi-particle thus also becomes "unstable". Of course, it's not necessarily unstable in the sense that it decays to other particles (which it cannot in $\phi^4$ theory in the vacuum due to energy-momentum conservation), but it simply means that through scattering with other particles in the medium it is scattered out of a given energy-momentum state. This is known as "collisional broadening".

Now in the many-body case the closed diagrams, i.e., diagrams without external legs, become much more important than in vacuum QFT, because in the latter case they just contribute to a phase factor, the vacuum-to-vacuum transition amplitude, which is canceled in all physical results anyway (that's proven in the sense that you only need to consider connected diagrams in vacuum physics, while all the unconnected diagrams cancel when taking the absolute-squared value to calculate the transition probabilities from the S-matrix elements).

Now the Wick theorem in QFT relates expectation values of time (or imaginary-time in thermal QFT in imaginary-time Matsubara formalism described above) ordered products wrt. to the given statistical operator to expectation values of normal-ordered products (which in the vacuum case simply cancel but are non-trivial in the finite-temperature/density case) and products of two-point Green's functions (also called "contractions" in this context). This is true if and only if the statistical operator is of the form $\exp(A)$, where $A$ is quadratic in the fields and their derivatives. This explains, why the Wick theorem is applicable in perturbation theory (both in the vacuum and at finite temperature/density): Here you use the free Hamiltonian to take expectation values, and your diagrams provide the corrections from the interactions as well as the correction to the statistical operator order by order in the coupling constant(s).

If you have more complicated statistical operators you use to take expectation values, Wick's theorem does not hold any longer, and you need a whole hierarchy of n-point functions (correlation functions). Then life becomes very complicated, and that's why this case usually is circumvented with clever tricks to map such a problem also to a usual one, where the Wick theorem holds (e.g., by switching on the correlations "adiabatically" in the remote past).

All this can also be derived by making use of the path-integral formalism and working with appropriate functionals. There Wick's theorem boils down to the algebra of functional derivatives of the generating functional $Z[J]$ of (disconnected) n-point Green's functions. For the free Hamiltonian it boils down to a Gaussian in $J$ with the coefficient given by the free-particle propagator.

For details of the relativistic many-body case, you can have a look at my lecture notes:

http://fias.uni-frankfurt.de/~hees/publ/off-eq-qft.pdf

A very good paper on the non-relativistic (non-equilibrium) case, using the operator formulation, where you also find a proof of the above mentioned generalized Wick theorem, is

Danielewicz, P.: Quantum Theory of Nonequilibrium Processes I, Ann. Phys. 152, 239, 1984
http://dx.doi.org/10.1016/0003-4916(84)90092-7

Another standard paper, using the path-integral formalism for the relativistic theory, is

Landsmann, N. P., van Weert, Ch. G.: Real- and Imaginary-time Field Theory at Finite Temperature and Density, Physics Reports 145, 141, 1987
http://dx.doi.org/10.1016/0370-1573(87)90121-9

On the case with initial correlations, see

Chou, K., Su, Z., Hao, B., Yu, Lu: Equilibrium and Nonequilibrium Formalisms made unified, Phys. Rept. 118, 1–131, 1985
http://dx.doi.org/10.1016/0370-1573(85)90136-X