QFT Signature: Does it Have its Own Unique Formula?

[Heirot]

By looking at some physical formula and noticing various symbols like c and hbar, one can immediately say that the formula in question describes relativistic / quantum effects and not classical. What about QFT? Is there any way one can, only by looking at the formula, say that it's a product of QFT formalism instead of standard (relativistic) quantum mechanics? I.e. does QFT has its own special signature?

Thanks

 

[Vanadium 50]

No. hbar is hbar, I'm afraid.

 

[Chopin]

Well, QFT is essentially the marriage of quantum mechanics with special relativity, so logically you'll see some signatures of both of those things in it. Specifically, since we're dealing with relativistic objects, you'll see lots of four-vectors and Lorentz-invariant expressions, and since we're also dealing with quantum mechanics, you'll see lots of bra-ket notation, operators, and commutators.

The most distinct thing about QFT expressions is probably that you see states being operated on with operator-valued fields, so you'll see lots of expressions like $\langle \psi|\phi(x)|\psi\rangle$, where $\phi(x)$ is an operator-valued field over $x$, which is a four-vector of position in space and time. That's different than non-relativistic quantum mechanics, where you'll see states being operated on, but either by fixed operators, or, at most, time-dependent operators like $H(t)|\psi\rangle$, but never a space-dependent operator. The spacetime dependence of the operators in QFT comes about because it's how you ensure locality, which is a relativity thing, so it's not something that comes up until you try to mix QM with relativity.

 

[vanhees71]

Of course, relativistic quantum theory is most conveniently expressed in form of quantum field theory, but quantum field theory is more general. In fact it's the most general formulation of quantum theory dealing with systems of particles whose number is not necessarily conserved. That's why it is particularly well suited for the relativistic theory since in this case only systems of free particles admit the definition of conserved particle numbers, while for interacting particles there's always the possibility to create new particles or destroy particles. There one has only charges as conserved particle-number like particles, but this is not really a particle number. E.g. electric charge is conserved, and this means one can always only create particle-antiparticle pairs, while the net-charge number is conserved.

In non-relativistic theory one very often has models with conserved particle number, and then quantum field theory is equivalent to quantum theory with a fixed particle number. Nevertheless also there quantum field theory can be very convenient to describe many-particle systems in and out of thermal equilibrium. As it turns out often one can describe such systems in terms of a quasiparticle picture, where collective modes of the system are described by a particle-like model. One example are lattice vibrations (sound waves) of solids, corresponding to quasiparticles called phonons. Then the behavior of the electrons within the solid can be described as interactions between these phonons (quantized Bose fields) and quantized fermion fields, which might also be "dressed" and have another mass than in the vacuum (heavy-fermion theory).

That's why I said that quantum field theory is a very general (if not the most general) scheme to describe quantum systems, including those in the relativistic and non-relativistic realm.