Time in relativity and quantum mechanics are completely different. The difference between them amounts to an obstruction.
Relativity defines time as an invariant. The clock on some path in spacetime counts off time in discrete equal intervals that count a proper time that measures the length of the spacetime path. For a particle in a general motion, say some wiggly path which is contained in a light cone at every point, the proper time is a sum over infinitesimal elements
ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2
and the proper time is a sum or integration over all these infinitesimal elements along the path
s = ∫ds.
It is important to recognize this as the real definition of time. The notion of time in the coordinate representation, what we call t, is a coordinate representation and not a physically invariant notion of time. One can transform to another coordinate frame with a different definition of this coordinate time, call it t', but the invariant interval = proper time s or ds is invariant. The coordinate time t is then just a label one imposes on spacetime, which spacetime ultimately does not care about; it is just our invention.
We will return to this, but now I discuss the meaning of time in quantum field theory. Quantum mechanics is a theory of waves which obey a differential equation for a wave. The scalar relativistic wave equation comes from the invariant momentum interval
(mc^2)^2 = E^2 - (pc)^2,
for a particle of mass m with momentum p and energy E. This is a momentum-energy version of the space plus time invariant interval or proper time above. Here the mass of the particle m is the invariant. The duality between spatial and momentum variables is a cornerstone of classical mechanics and Fourier transforms in wave mechanics. I will from now on set c = 1. The quantization procedure is to define the momentum and energy as the operators
p = -iħ∇ = -iħ(i∂_x j∂_y k∂_z)
E = -iħ∂_t
for ħ the Planck unit of action (which I now set to one). These operators act on a wave function ψ = ψ(r, t) and the differential wave equation is
(∇^2 - ∂_t^2)ψ = m^2ψ.
We now just consider this wave equation. It is set up with some initial field configuration, eg initial data, which is established on some spatial surface of three dimensions. This means that one must fix a time, or time slice, according to the coordinate time where one fixes initial data. The subsequent evolution of the wave is according to this variable t, seen in the differential part ∂_t in the wave equation above. This means we have evolution of quantum waves, which are determined in a second quantization procedure by quantum field, according to a coordinate time --- not the invariant proper time of relativity. The quantum concept of time then demands some time slicing of spacetime or coordinate condition, which is not a fundamental observable in relativity.
The clock measures proper time by traveling on a spacetime path marks off time according to some mechanism. We think of the clock as being a real clock, with gears, springs or some atomic system, maybe vibrating nuclei, cavorting quarks etc, which obeys some set of physical rules. Let us not think of the clock as some mathematical idealization, but as some sort of physical contrivance. The clock then functions according to some physics, such as the quantum field theory above. This is particularly the case if the clock is some sort of atomic clock or that depends upon some quantum periodicity. Consequently the proper time in physical relativity is not purely a mathematical idealization used in relativity theory.
The infinitesimal proper time parameterized by some arbitrary λ is such that
∫ds = ∫(ds/dλ)dλ
so that L = ds/dλ is a Lagrangian. We know from classical mechanics the Lagrangian may be written according to a Hamiltonian so that
Ldλ = π^{ij}dg_{ij} - Hdλ,
where the Hamiltonian H = H(π, g, ψ, ∂_iψ) is a function of the metric and some quantum field. This is the constraint which fixes the action on a contact manifold of solution. The π^{ij} is a momentum conjugate of the metric of space. This then runs into further problems if the wave function(al) is dependent on the metric. This Hamiltonian acts on the wave function(al) HΨ[g, ψ] = 0 with no time reference. The quantum mechanics prevents any use of a particular time slice necessary for quantum field theory.
The two concepts of time are intertwined in some manner, but as yet there are fundamental obstructions which prevent a consistent description..
Cheers LC