What are quantum fields mathematically?

The definition of a quantum field depends slightly on the formalism that you adopt, but globally, quantum fields are defined as operator-valued distributions. That is, if you have a quantum field $\Phi$, it is defined as

$$\Phi : \mathscr D(\mathcal M) \to\mathcal B(\mathscr H)$$

It maps smooth functions with compact support on the spacetime manifold to linear operators on the Hilbert space where your quantum theory is defined. By some abuse of notation, we sometimes write it as $\Phi(x)$, although this is only well defined if the distribution is also itself a smooth function.

This has some difficulties associated to it (since distributions can't be multiplied together easily, and QFT involves a lot of products of fields), meaning one has to use such methods as wavefront sets and renormalizations to make sense of everything.

There is no mathematically sound formulation of realistic QFT yet so at this point we have no real answer to your question. The QFT that physicists use to make predictions is in the so-called Lagrangian formulation, which is a heuristic framework for obtaining perturbative expansions using Feynman diagrams. There is also algebraic or axiomatic QFT, mathematically well-defined but so far confined to free theories and toy models. The idea is that QFT must satisfy a list of axioms, the Wightman axioms being most commonly used, and the challenge is to construct realistic theories that satisfy them. Mathematically constructing a Yang-Mills theory with a mass gap is one of the Millenium problems.

In algebraic QFT fields are identified with operator-valued distributions, and the Fock space picture is a dual representation of them. This duality is similar to the Schrödinger vs Heisenberg pictures in quantum mechanics. The idea is that the Hilbert space of quantum fields, as distributions associated to localized regions of spacetime, is unitarily equivalent to the Fock space, where creation and annihilation operators are defined, and which is much more commonly used in practice. That is the Fock space of second quantization, so those operators are not the same as the field operators, which are quantized versions of classical fields (intuitively, the Fock space operators are "global" whereas the field operators are "localized"):

"Fortunately, the operators on a QFT Hilbert space include a set of field operators. If a particular wave equation is satisfied by a classical field $\phi(x)$, it will also be satisfied in operator equation form by a set of operators $\widehat{\phi}(x)$ on the state space of the quantized version of the field theory. Speaking somewhat imprecisely, $\widehat{\phi}(x)$ acts like a field of operators, assigning to each point x an operator with expectation value $(\psi,\widehat{\phi}(x)\psi)$. As the state evolves dynamically, these expectation values will evolve like the values of a classical field. The set of field operators is sometimes called the operator-valued quantum field. One caveat that will be important later: Strictly speaking, we cannot construct a nontrivial field of operators $\widehat{\phi}(x)$ defined at points. But it is possible to define a “smeared” quantum field by convolution with test functions.

[...] We need an interpretation of field-theoretic states to determine which physically contingent facts they represent. In single-particle QM, a state is a superposition of states with determinate values for the theory’s observables (e.g. position and momentum)... in field theories we’re interested in systems that take on values for some field $\phi(x)$ and its conjugate momentum $\pi(x)$. So, when quantizing a field theory, we should just do to the field what we did to the mechanical system to generate QM. Impose commutation relations on $\phi(x)$ and $\pi(x)$, and move our states to the Hilbert space of wavefunctionals ($\Psi(\phi)$) that describe superpositions of different classical field configurations.

The equivalence to Fock space picture can be proved for free QFT, but axiomatic QFT has difficulties incorporating interactions or defining position operators. Because of this some argue that neither quantum field nor Fock space/particle interpretations can survive in a mathematically mature QFT, see e.g. Baker's Against Field Interpretations of Quantum Field Theory, from which the above quote is taken.

Wallace has a nice review In defence of naivete: The conceptual status of Lagrangian QFT that analyzes the mathematical structure of QFT as it is practiced, and argues to the contrary that it can be seen as a valid approximation of what algebraic QFT may one day yield. If that is the case then operator-valued distributions and Fock space states, interpreted as particle states, will be effective realizations of what quantum fields "are" at low energy levels:

"We have argued that such QFTs can be made into perfectly well-defined quantum theories provided we take the high-energy cutoff absolutely seriously; that the multiple ways of doing this are not in conflict provided that we understand them as approximations to the structure of some deeper, as yet unknown theory; that the existence of inequivalent representations is not a problem; that a concept of localisation can be defined for such theories which is adequate to analyse at least some of the practical problems with which we are confronted; and that the inexactness inherent in that concept is neither unique to relativistic quantum mechanics, nor in any way problematic.

The replies which suggest that the answer to "What is a quantum field?" is unclear or even open are wrong.

The impression that this could be unclear is owed to the standard textbooks sticking to the heuristics that helped Tomonaga-Schwinger-Feynman-Dyson to guess the theory many decades back, but the mathematical nature of realistic quantum field theory was completely understood by the mid 70s and further developed since. A survey of the state of the art is at

• PhysicsFormums-Insights -- Introduction to perturbative quantum field theory

First of all it is worthwhile to realize that there is a difference between a field configuration and an observable on the space of all field configurations.

A field itself, either in classical physics or in its quantization, is simply a function on spacetime, assigning to each spacetime point the "value" of that field at that point. Or rather, more generally it is a section of a bundle over spacetime, called the field bundle. For instance if the field bundle is a spin bundle then the field is a spinor, if it is the differential form bundle, then the field is a gauge potential as for electromagnetism, etc.

Now from the Lagrangian density one obtains two things: the equations of motion as well as a pre-symplectic form on the space of all those field histories which solve the equations of motion. This is called the covariant phase space of the theory.

An observable is a function on this covariant phase space. It sends any field history to a number, the "value of that observable on that field history". But since the covariant phase space is itself a space of functions (or rather sections), a function on it is a functional.

Among these are the "point evaluation functionals", i.e. the observables whose value on a field history is the value of that field at a given point. The business about distributions is simply that on these point evaluation functionals the Peierls-Poisson bracket is not defined (only its integral kernel is defined, which is what you see in the textbooks). So one restricts to those observables which are functionals on the space of field histories on which the Poisson bracket actually closes. These are smearings of the point evaluation functionals by compactly supported spacetime functions. So then a point evaluation functional becomes a map that once a smearing function has been specified yields an observable. This way already classical point evaluation field observables are distributions: "classical observable-valued distributions".

Now all that happens in quantization, is that the pointwise product algebra of functionals on the covariant phase space gets deformed to a non-commutative algebra. It is traditional to demand to represent this algebra inside an algebra of operators on a Hilbert space, but for the most part this is a red herring. What counts is the non-commutative algebra of quantum observables. For computing the predictions of the theory, its scattering amplitudes, it is not actually necessary to represent this by operator algebra.

Anyway, whether you like to represent the non-commutative algebra of quantum observables by operators or not, in any case the result now is that a point evaluation functional is something that reads in a smearing function and then produces the corresponding observable, exhibited now as an element of a non-commutative algebra. In this way quantum observables on fields are algebra-element valued (e.g. operator-algebra element valued) distributions.

And, yes, for free fields this does yield the familiar creation annihilation operators, for details on how this works see

• on the nLab at Wick algebra.

There is detailed exposition of these questions at

• geometry of physics -- A first idea of quantum field theory

Presently this is written up to the classical story. For the quantum theory check out the site again in two months from now.