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  "Classical" Lagrangian formulation of quantum field theory

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Are there some articles or books in which Quantum Field Theory is treated systematically using a least action principle? I don't mean the path integral formalism, but a "true", "classical" Lagrangian formulation, and the dynamics being obtained by minimizing the action.

For example, the Klein-Gordon and Dirac equations have Lagrangian formulations, as well as the nonrelativistic Schrödinger equation, including for more particles, for example for two particles we have the following Lagrangian

\[\mathcal L(x_1,x_2,t)=\hbar Im\{\psi^\ast\frac{\partial \psi}{\partial t}\}+\frac{\hbar^2}{2m_1}\nabla_1\psi^\ast\cdot\nabla_1\psi + \frac{\hbar^2}{2m_2}\nabla_2\psi^\ast\cdot\nabla_2\psi + V(x_1,x_2)\psi^\ast\psi\]

I imagine that we can include more particles, and linear combinations of states of different numbers of particles, so I see no reason why the same can't work for a variable number of particles. Also, QFT being a dynamical system, quantum as it is, I still see no reason why it can't be described as minimizing an action.

But in all references I found, including by searching online, one moves to antiparticles and creation and annihilation operators, and usually what's called Lagrangian or variational formulation is actually Feynman's path integral approach, which uses the action in a different way than the least action principle. I am also aware of the question https://physicsoverflow.org/20612/quantum-mechanics-as-classical-field-theory?show=20612 and the answers, but unless I missed something for the references from the answers, they are all about path integrals. Also, the equation written above is from this article http://www.scottforth.com/publications/2002_AmJPhys_Styer%20et%20al.pdf section G, and they give as reference for QFT Itzykson and Zuber's Quantum Field Theory, yet there again I find the path integral formulation.

Maybe it doesn't work and it turns out that the so-called second quantization is needed, and some probabilistic interpretation, but I wonder if there's some literature in which it is treated systematically, and shown why is or is not equivalent to the usual formulations.

asked Nov 18 in Theoretical Physics by Cristi Stoica (275 points) [ revision history ]
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There is a Schwinger quantum least action principle with no path intergals.

@New Student Some of these problems and others bother me too. Do you know a formulation of QFT free of problems?

See the free access book "Between classical and quantum mechanics" by N.P. Landsman, chapter 5 and 6 and all their links, particularly those about quantum mean-field systems. Perhaps also a more general approach with P. Cardaliaguet "Notes on mean-field systems". In some cases, it is easy to derive a probabilist LAP from its efficiency at the macro level as a convenient method. The challenge is to find a quantum theorem, ie a derivation from the first principles only.

Thanks, @igael. On P. Cardaliaguet's page I see Notes on Mean Field Games, I take it as this contains the more general approach.

Weinberg's book on QFT derives the Feynman rules for S-matrix elements of QED before introducing the path integral.

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@Vladimir Kalitvianski Schwinger's seemed to me too pretty close to what I want at some point, but it's not close enough, because the action is an operator, so "least" is not quite defined in this case, but at least the action operator is stationary (constant), so it seems closer than the path integral approach to a variational principle indeed.

3 Answers

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If one wishes to compare QFT with classical physics, using a formalism common to them both, my feeling is that the classical Lagrangian formalism is not immediately appropriate because it is not a probabilistic formalism. That is, one could construct a Lagrangian for a deterministic classical field theory, but one would then still have to introduce a probability density over the initial conditions, a Liouville state over the phase space of the field theory, which would presumably be something like a Gibbs distribution, in terms of a classical field $\phi(x)$ and its phase space conjugate $\pi(x)$, $\exp(-\beta H(\phi(x),\pi(x)))$, but to equate to QFT the Liouville state would have to be Lorentz invariant, which the Gibbs state is not. The path integral formalism is a generating function for the time-ordered Vacuum Expectation Values, which makes QFT look classical because the algebra of operators under time-ordering is commutative, however this discards significant information contained in the noncommutative algebra.

If you want a Lagrangian formalism and only a Lagrangian formalism, the rest of my answer won't satisfy you, but I have found it more helpful to consider classical physics in a Koopman-von Neumann formalism (which dates from a paper in 1931 in which Koopman presented classical physics as an operator algebra acting on a Hilbert space), so that one is working with a Hilbert space presentation of both random fields and quantum fields. In this shared formalism, one can construct isomorphisms relatively easily, which makes a welcome change from the awkwardness of the correspondence principle. I develop this approach for field theories in arXiv:1709.06711, however I encourage you to look at the version I submitted to Physica Scripta two weeks ago, https://www.dropbox.com/s/xoj3gt81h2fonmf/1709.06711-asSubmittedTo-PhysicaScripta.pdf?dl=0, because this version makes contact with the Tsang&Caves paper "Evading quantum mechanics" from 2012, as well as making some other smaller changes.

Within the Koopman-von Neumann formalism, there is, crucially, a natural introduction of noncommutative algebraic structure because of the Poisson bracket. I presume that one could construct a Lagrangian formalism following something like the pattern I've constructed in the paper I cite above, however taking powers of operator-valued distributions of random or quantum field theories is a fraught business that IMO is best avoided.

answered Nov 18 by Peter Morgan (1,200 points) [ no revision ]

For the time being, I would be happy even without a probabilistic interpretation, just fields. I'll look into the references, thanks!

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I do not know any book on Quantum Field Theory (QFT) which uses the least action principle instead of the path integral. The likely reason is that QFT in the continuum cannot be done in this way. Here is my argument:

You wrote "and the dynamics being obtained by minimizing the action". The question is: the dynamics of what? The least action principle gives you an equation of motion for a quantity which is usually called "field". In quantum mechanics, the relevant equation of motion is the Schrödinger equation, $\partial_t \psi = H\psi$. Once we know this result, we can actually ignore the principle that gave it to us; but we still have to clarify what $\psi$ is.

The naive picture of QFT is that $\psi$ is a state in a Hilbert space, the Fock space. This actually works in lattice QFT, where the Hilbert space is a finite-dimensional space (for spins and fermions, at least). However, in the continuum, it is a popular illusion. The problem is that any QFT in the continuum has to be regularized, and this regularization may unexpectedly break symmetries; this phenomenon is known as a quantum anomaly (This can give rise to very powerful constraints: For example, the quark charges are fixed to multiples of 1/3e by the requirement that Quantum Chromodynamics has to be free of quantum anomalies). Moreover, the regularization is usually note unique, but depends on some arbitrary quantity, the cut-off; and one has to invoke renormalization to argue that the choice of cut-off is not important for the low energy physics. The combination of these to facts make it hard to impossible to define a satisfying Hilbert space for interacting QFTs, but here I have to admit that I am not familiar enough with the attempts.

You seem to be dismayed by the path integral approach, but it has the benefit that it can deal with quantum anomalies. The confusion is probably that in this approach, the "field" is not an actual quantity that evolves in time; it is merely a helper quantity with no dynamics of its own, it is just something to integrate over. (That said, it does label coherent states, and can be used as a variable to describe the dynamics in a semiclassical regime, so it is not that far removed from being a dynamical field.) But given that an "equation of motion" approach to QFT does not quite exist, I think it is a small price to pay.

answered Nov 26 by Greg Graviton (775 points) [ no revision ]
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I do not care here if you stone me for my answer, but even in the Classical Mechanics (CM) the least action principle is kind of cheating: one specifies two positions $x(t_1)$ and $x(t_2)$ in order to "derive" the equations, but one never uses $x(t_2)$ in practical calculations because the future positions are unknown. Instead, one contents oneself with the equations (admitting so many quite different stationary trajectories) and the initial data to find the unique future position. In other words, the least action thing serves just to "obtain" already obtained empirical equations and nothing else, let us not fool ourself. The equations do not determine a specific evolution!

In QFT the equations of motion are also postulated, by analogy with the classical empirical postulating. It is possible to "derive" them from some least action principle, but this is again some sort of cheating. We always solve an initial data problem rather than a "brachistochrone" variational problem or a boundary condition problem. Yes, in QM this principle is burdened with a wavy nature of the whole picture. Thus either you use the Schwinger operator approach, or the path integral formulation, and this differs QM from CM.

Greg is right by saying that there are also regularization and renormalization problems, which burden the understanding in the QFT, but I believe (read my toy problems) it just underlines our inability to approach creatively the Nature description rather than a true drawback of the least action principle in QFT.

You see, even with this "least action principle" we must modify the equation solutions because without modifying they are good for nothing. As R. Feynman famously said, guessing the right equation (from observed physics) is still on our table. The least action principle guides us to nothing new and it does not guarantee problem-free (=physical) solutions.

answered Nov 26 by Vladimir Kalitvianski (97 points) [ revision history ]
edited Nov 27 by Vladimir Kalitvianski

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