Reconciling two types of time in QFT

Question

When I took field theory 25 years ago, I learned to do the mathematical manipulations very fluently but had no idea whatsoever what it all meant. Now, with a more mature understanding of quantum mechanics, the following issue confuses me.

My understanding at this point is that QFT is just quantum mechanics with fields as the degrees of freedom. In nonrelativistic QM, we have questions like "where is the particle?," and these are answered by observables like a position operator. In relativistic QM, we ask "what is the electric field at this point in spacetime?," and there is an observable for that. The structure of QM, which is shared by QFT, is that we have a Hilbert space and there's unitary time evolution in which the hamiltonian is the generator of time translations. (Let's say we're using the Schrödinger picture.) Time is a parameter, not an observable.

So if I just go ahead and see how this should work out according to these principles, I would expect that I have a whole bunch of degrees of freedom $\phi_{x,t}$, where $\phi$ is the field and $(x,t)$ are Minkowski coordinates. Let's say I'm using a lattice in a box with $m$ spacetime points. A state in the Hilbert space is a function from $\mathbb{R}^m$ to $\mathbb{C}$ that takes some configuration $(\phi_{x_1,t_1},\phi_{x_2,t_2},\ldots)$ as an input and gives back a complex number as an output. I can then do a unitary operation $\exp(i\lambda H)$ and time-evolve the system from one value of the time coordinate $\lambda$ to another.

So here I have two notions of time that seem completely different. (1) I have the Minkowski coordinate $t$, which is specific to a certain observer's frame of reference, is tied in to the structure of spacetime through the metric $\eta$, and presumably enters into the hamiltonian when we specify how $\phi_{x_i,t_i}$ couples to $\phi_{x_j,t_j}$. (2) I have the quantum-mechanical time parameter, which I've notated $\lambda$. It has nothing to do with any observer's frame of reference, and is not tied in to the structure of spacetime through the metric.

There seems to be nothing wrong with this setup in principle, and it seems to be the canonical thing you get when you combine the principles of special relativity with the principles of quantum mechanics. And yet it has these two different notions of time in it that don't seem to tie together in the way we would normally expect based on how we experience time in experiments. How are these two notions of time reconciled?

Of course we could get into things like EPR, but this seems to me like something much more basic that I would like to understand before I can even coherently state what EPR is about. The story-line for EPR is sort of like, "Oh no, collapse is instantaneous, so doesn't that violate causality?" But we can't even state that as more than a non sequitur until the $\lambda$-vs-$t$ issue is resolved, because "instantaneous" refers to $\lambda$, while "causality" has to do with $t$, which seems unrelated.

Answer 1

Let $M^4$ denote the four dimensional Minkowski spacetime. $$\Phi(x)$$ is the quantum field at the event $x\in M^4$ and here $x$ is nothing but a label.

Next there is the action of the (proper orthochronous) Poincaré group $\cal P$ in terms of isometries on $M^4$. If we fix arbitrarily a Minkowskian reference frame we can represent the Poincaré group with respect to these axes. Since the action of $\cal P$ on $M^4$ is transitive, the choice of a Minkowskian reference frame is irrelevant.

Let $p_0,p_1,p_2,p_3$ be the generators of spacetime translations along preferred Minkowskian axes. If $n$ is a timelike unit-vector future-pointing $n^\mu p_\mu$ is the generator of time displacements along $n$ (so a generic direction in the future light cone). The action in $M^4$ of the generated one-parameter group is $$\exp(\tau n^\mu p_\mu) x = x + \tau n$$ The action of the isometry group $\cal P$ can be implemented in the Hilbert space through a strongly continuous unitary representation $${\cal P} \ni g \mapsto U(g)$$ In particular whe have the representation of the subgroup $$U(\exp(\tau n^\mu p_\mu))= e^{-i \tau n^\mu P_\mu}\:,$$ where I have introduced the selfadjoint operator $n^\mu P_\mu$ representing (minus) the Hamiltonian operator associated to the temporal direction $n$. The action on the quantum fields is $$U(g) \Phi(x) U(g)^\dagger = \Phi(g(x))$$ In the special case of time translations, $$U(\exp(\tau n^\mu p_\mu)) \Phi(x) U(\exp(\tau n^\mu p_\mu))^\dagger = \Phi(x+ \tau n)\:.$$ You see that the parameter $\tau$ associated to the time quantum translation acts as as an isometry on the manifold: moving the labels of the points. Time evolution in Heisenberg picture is the inverse transformation of time trnaslation.

All that makes sense just because we have a coordinate system made of curves tangent to Killing fields of the manifolds. In curved spacetime, in general, this picture does not exist. In case a timelike Killing field exists a similar construction is still possible.

In general in QFT in Minkowski spacetime a safe point of view is Heisenberg picture, extending this picture to include the action of the full $\cal P$ not only time-translation (the inverse of time evolution).

EPR refers to correlations between regions of spacetime which are causally separated. So, we are dealing with "labels" and we are assuming that "$x$" and "$x'$" are such that there is no future-directed causal curve joining them. Here quantum time evolution does not enter the game. What happens is that outcomes of quantum measurments at $x$ and $x'$ on an entagled system turn out to be show correlations (...).

Answer 2

The construction you described is, indeed, possible, but this is not the standard way to do QFT. In standard wave functional approach, we work in some specific (but arbitrary) reference frame. In this reference frame we chacterize basis field state as a function of space variables only: $\phi_x$. And in this reference frame we have unitary evolution $\exp(itH)$.

The theory constructed in this way contains only one time ($t$), but such theory is not explicitly Lorentz invariant. But in fact it is Lorentz invariant implicitly — if the Hamiltonian is correct — and this can be verified.

To construct an explicitly Lorentz invariant theory, other approaches are used. The approach that you described is called "parameterized QFT" and it is very rare. More commonly used approaches are second quantization and path integral quantization, and these approaches also contain only one time variable.