In, QFT, do the excitations in the quantum fields exist physically? In QFT, all particles are really just excitations in their quantum fields, and we know that these fields are just mathematical.For example, an electron is an excitation of the electron field. But my question is, is the excitation of the field (the particle) mathematical too, or is it physically real? And if the excitations of the fields do not really physically exist, does that mean that reality doesn't physically exist either?
 A: This is an experimentalist's view.
Here is a definition of a field in physics:

In physics, a field is a physical quantity that has a value for each point in space and time

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A field can be classified as a scalar field, a vector field, a spinor field or a tensor field according to whether the represented physical quantity is a scalar, a vector, a spinor or a tensor, respectively.

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Moreover, within each category (scalar, vector, tensor), a field can be either a classical field or a quantum field, depending on whether it is characterized by numbers or quantum operators respectively. In fact in this theory an equivalent representation of field is a field particle, namely a boson.

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You state:

For example, an electron is an excitation of the electron field.

Take an electron describing a trajectory in space time. At the (x,y,z,t) point a creation operator creates it and at (x+delta(x), y+delta(y),z +delta(z) t+delta(t)) destroys it to be picked up by the next (x',y',z',t') creation .... The functional form of the trajectory is described by the solution of the appropriate quantum mechanical solution of the boundary condition problem.
Having disposed of the luminiferous aether QFT manages to create an underlying structure where excitations propagate defining particles. The difference lies in that Lorenz invariance holds for these fields.

But my question is, is the excitation of the field (the particle) mathematical too, or is it physically real? And if the excitations of the fields do not really physically exist, does that mean that reality doesn't physically exist either?

The particles are real in the sense they have macroscopic effects and are measured in our laboratories.
The mathematics of field theory is real, as real as the parabolas predicted for throwing a ball in a gravitational field.
QFT is a mathematical model to deal with the  many body problem of quantum mechanics
A: So called "particle states" (momentum eigenstates) in QFT are not part of the Hilbert space (they are not normalizable), but only the rigged Hilbert space. As such, they cannot actually be physical states. Real states corresponding to real particles are more likely to be somewhat localized in both position and momentum.
The particle states are very useful as a basis for wavefunctions, though, similar to sine waves in a Fourier transform.
A: Properties of the quantum fields (its mass density, charge density, response to external fields, etc.) can be measured and predicted in the same way as all quantum observables. There is therefore nothing unreal about a quantum field. They are at least as real as their excitations, the elementary particles.
In fact, quantum fields are far more physical than mathematical. In particular, interacting quantum fields in 4 dimensions do not yet make mathematical sense, while physicists use them all the time.
The field is what really exists, i.e., the medium, and the excitations are its oscillations. Just like water waves are excitations (local, extended oscillations) of water, which is the medium carrying the waves. The main difference is that water waves are not quantized, so that there are no 'elementary' excitations.
