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I am reviewing some basic aspects of the passage from classical field theory to quantum field theory. The steps to obtain a quantum theory are

  • (I). Find a suitable classical action $S$ whose minimisation yields classical equations of motion.
  • (II). Obtain from $S$ the Lagrangian $L$.
  • (III). Find the conjugate momenta $\Pi_a$ of the relevant fields $\phi^a$ as $\Pi_a\equiv\frac{\delta L}{\delta \dot\phi^{a}}$.
  • (IV). Construct the Hamiltonian via a Legendre transformation.
  • (V). Replace the fields and momenta by operators and upgrade the canonical Poisson brackets $\{\phi^a(x),\;\Pi_a(y)\}_{\rm P.B.}$ to commutators $[\hat{\phi}^a(x),\;\hat\Pi_a(y)]$.

Now, what happens if I want to promote just some of the fields to quantum fields but replacing others as quantum. For instance, imagine that I have two scalar fields $\phi(x)$, $\varphi(x)$. I want a theory in which $\phi(x)$ is described as a classical field and $\varphi(x)\to\hat\varphi(x)$ is promoted to a quantum field. Is there a direct way to do this in step (V)? Should I retain the Poisson bracket structure for observables that only depend on $\phi(x)$ and use the commutator structure for observables depending only on $\hat\varphi(x)$? What happens for observables depending on both fields?

TopoLynch
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3 Answers3

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It sort of depends on what you want to do. Say you want to keep $\phi$ as classical and replace $\varphi$ by an operator. Do you want to consider $\phi$ as dynamical? In other words, do you want to impose equations of motion on it? If so, this can be problematic, since an equation of the form $$ \partial^2\phi=V'(\phi,\varphi) $$ doesn't make sense if $\phi$ is classical and $\varphi$ is quantum. Indeed, the r.h.s. is clearly quantum, and therefore so must the l.h.s.

If $\phi$ and $\varphi$ are decoupled then yes, you can simply replace one by an operator, and keep the other classical. Of course, this is valid but boring.

In general, the only way to mix classical and quantum operators is to impose equations of motion on the latter, but not on the former. In other words, we let the classical fields be fixed functions, that we choose by hand, and let the quantum fields be dynamical, with equations of motion that depend on the classical fields. This is a very common situation, usually referred to as "coupling the theory to a background". A background, in this context, is simply a classical field that we choose by hand. This is a very useful tool in QFT.

Once you declare that some classical fields are fixed, it is meaningless to talk about their Poisson brackets. The only structure that you have is commutators for the quantum fields, and these are oblivious to the classical ones -- operators commute with c-numbers.

AccidentalFourierTransform
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OP seems to essentially ponder if hybrid models of quantum and classical fields exist?

  • Feynman argued (around 1957) that if a theory has a quantum field, then it forces other fields to be quantum fields as well. In particular, Feynman's argument is used to claim that gravity should be quantized, cf. e.g. EOS & this, this, this Phys.SE posts.

  • Recently (since 2023) J. Oppenheim et al have explored loopholes in Feynman's argument, cf. e.g. ScienceNews, quantamagazine & this Phys.SE post.

Qmechanic
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Notwithstanding the already good answer of @AccidentalFourierTransform, I want to add some remarks about the dynamics of the classical counterpart in a semiclassical analysis. There is a way that one can allow such a classical field to evolve.

The scenario is where the field being modeled classically is strong. If we consider such a scenario on phase space, we see that such a field is located far from the origin of phase space. The evolution of the states in such a system can be described by a Fokker-Planck type of equation. Then we can redefine the phase space variable for this strong field relative to the location of its state. The leading order in the expansion in terms of the new variable then represents the semi-classical approximation in which the classical field is fixed. Sub-leading orders provide relationships that can be used to obtain equations for the dynamics of the state of the classical field.

flippiefanus
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