A quantum field is an operator valued function, that is, a function $\varphi(x)$ defined on spacetime which assigns operators on a Hilbert space to each event $x$. In a more rigorous approach a quantum field could be defined as an operator valued distribution on spacetime.

Anyway, it is quite common that these quantum fields obey differential equations, like the Klein-Gordon equation $$(\Box +m^2)\varphi=0$$ and Dirac's equation $$(i\gamma^\mu \partial_\mu - m)\psi=0.$$

In that sense we need to understand what is the derivative of a quantum field. In this seems a little complicated.

Of course one can say: "a quantum field takes values in a Hilbert space so you can use the Frechet derivative", but it is not even clear what Hilbert space it is that the quantum fields takes values on. Also, as is clear from Quantum Mechanics, most of the operators we deal in QM are unbounded and hence discontinuous. I believe this would have a great impact on how should we deal with things like derivatives.

So, in order for quantum fields to satisfy differential equations, how is the correct way to define and understand the derivative of a quantum field? How can we make sense of quantum field differential equations?

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    $\begingroup$ I don't know about quantum field theory specifically, but in mathematics we often want to differentiate a function that isn't really differentiable, and we have a general framework of "weak derivatives" for how to do so. $\endgroup$
    – Ian
    Mar 20 '17 at 11:51

The fields of a QFT are not functions of the spatial coordinates $\boldsymbol x\in\mathbb R^n$, but operator-valued distributions (borrowing Wightman's terminology). The notion of the fields being functions of time ("sharp-time fields") can be kept in general, but their dependence on $\boldsymbol x$ is "too singular", so that they become distributions; fields need be smeared out in space.

Therefore we should write $\phi[f]$ instead of $\phi(\boldsymbol x)$, in the same way we should write $\delta[f]$ for the Dirac delta. In this sense, the commutation relations $$ [\phi(\boldsymbol x),\pi(\boldsymbol y)]=\delta(\boldsymbol x-\boldsymbol y) $$ should be written as $$ [\phi(f),\pi(g)]=(f,g) $$ for a certain scalar product $(\cdot,\cdot)$ on your space of test functions (that depends on the spin of $\phi$).

Similarly, the field equations $$ \dot\pi(\boldsymbol x)-\Delta \phi(\boldsymbol x)+m^2\phi(\boldsymbol x)=0 $$ should actually be written as $$ \dot\pi[f]-\phi[\Delta f]+m^2\phi[f]=0 $$

More generally, the naïve field equations $$ \mathscr D\phi(x)=0 $$ are nothing but a short-hand notation for $$ \phi[\mathscr Df]=0 $$ for all $f$ in the domain of $\phi$. Therefore, the derivatives acting on fields are to be understood in the sense of distributional derivatives (if $T$ is a distribution, then we define $T'[f]\equiv -T[f']$, etc.).

For free theories, the whole framework of operator-valued distributions is perfectly well understood, and one may work with all mathematical rigour one may wish. For interacting theories though, we are far from a mathematically sound theory.

For more details see for example On Relativistic Irreducible Quantum Fields Fulfilling CCR or Haag's theorem in renormalised quantum field theories. Also, anything by Wightman (e.g., PCT, Spin and Statistics, and All That).

  • $\begingroup$ Thanks @AccidentalFourierTransform. That's a quite nice approach, since this shifts the derivative from the quantum field to the functions on which it acts. There's however a question I have: distributions are perfectly well defined in flat-spacetime where we have the Schwartz space available. But what about curved spacetime? There we don't have the Schwartz space available. What becomes of quantum fields then? $\endgroup$
    – Gold
    Mar 26 '17 at 21:30
  • $\begingroup$ @user1620696 I'm afraid that I don't really know much about QFT in curved-space times. As far as I know, there is no straightforward generalisation of the Wightman axioms to curved manifolds, so in principle it is far from obvious how would one try to approach rigorous treatments of QFT other than in Minkowski. I think it is an open problem for the most part. I am not familiar with any serious attempt to construct operator-valued distributions in curved spacetime (but I'm sure people must have studied this in detail). $\endgroup$ Mar 26 '17 at 21:41
  • $\begingroup$ I just found the paper 1401.2026 which seems to address exactly what you asked. See also 0907.0416. Wald has got you covered! $\endgroup$ Mar 26 '17 at 21:45
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    $\begingroup$ @AccidentalFourierTransform : regarding construction of distributions over a curved spacetime, taking local maps on that curved spacetime allows you to define local distributions, and as the most common Schwarz space uses test function with compact support, don't we have there all what is needed for such a construction? BTW, I think you should make some efforts in your notation: "we should write $\phi[f]$ instead of $\phi(\boldsymbol x)$" doesn't make any sense, apart to who knows already what you could have meant. $\endgroup$
    – user130529
    Mar 28 '17 at 20:24
  • $\begingroup$ +1. Hello; if I may ask you for some reference on the operator-valued distribution nature of the quantum field operators. Maybe a reference for an even more basic analysis of the need for understanding quantum fields in such a way; the basic QFT notes I possess(Srednicki, Greiner) do not analyze the subject beyond the basic functional nature of the Lagrangian. Thank you. $\endgroup$ Apr 3 '17 at 7:39

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