# How to deal with entanglement using quantum relativistic equations?

When discussing relativistic quantum mechanics, the single particle picture is dropped completely which makes the description of just two entangled particles difficult. However when discussing locality and causality, relativistic quantum mechanics should provide some insight.

However can some description be built from Dirac's equation or relativistic quantum field theory?

• This is an interesting issue to consider. However, I see two problems with this post. First, it is unclear what the actual question is. Second, it has probably greater chances of getting a good answer on Physics SE. Also, I am afraid you may be disappointed with how things turn out. Specifically, I believe you'll find out that quantum field theory is necessary to answer your question. Sep 11, 2021 at 16:07
• Roughly, it goes like this. Single-particle QM is inconsistent with SR, e.g. QM predicts non-zero probability of finding a particle outside its lightcone. A central issue is that Schrödinger equation is derived from linear dispersion relation of non-relativistic CM. It turns out that obtaining a quantum counterpart to relativistic dispersion relation is tricky and once you have it you find that it has additional solutions corresponding to anti-particles. This enables pair production which means you cannot naively assume a fixed number of particles. And so you're led to quantum fields... Sep 11, 2021 at 16:19
• Can one come with a quantum field description of entanglement then? Sep 11, 2021 at 16:58
• Yes, see e.g. this note. Sep 11, 2021 at 17:02
• @AdamZalcman thanks, if you write an answer summarizing the idea I would gladly accept it Sep 11, 2021 at 17:26

## The need for quantum fields

There are two seemingly unrelated conceptual steps between quantum mechanics (QM) and quantum field theory (QFT). One step reconciles QM with special relativity (SR) and the other replaces a fixed finite number of particles with fields that have infinite degrees of freedom. These two steps appear unrelated, but in order to arrive at a consistent theory both are needed.

SR postulates that all laws of physics are invariant under Lorentz transformations. QM violates this principle, because Schrödinger equation, being derived from the non-relativistic dispersion relation, is not Lorentz invariant. The solution is to derive the quantum evolution equation from the relativistic dispersion relation. This can be done in two ways with one leading to Klein-Gordon equation and the other to Dirac equation.

Now, among the consequences of the quantum theory equipped with Lorentz invariant evolution is the existence of antimatter and pair production. Therefore, it is no longer consistent to assume that the number of particles, and degrees of freedom, is fixed. The upshot is that a consistent quantum theory that incorporates SR generally looks like a QFT.

## Dirac notation

It is not true that Dirac notation is not employed in QFT. However, unlike QM, QFT does not yet have a canonical mathematical formulation, so the formalism used may differ depending on the source. That said, Dirac notation is often not the most convenient. See creation and annihilation operators for a simple and widely used alternative.

## Entanglement

Entanglement is certainly present in quantum fields. In fact, spatially adjacent modes of a quantum field are known to be very strongly entangled, see this paper for details and a comprehensive discussion of entanglement in QFT. See also this paper for an example of a calculation of von Neumann entropy in a quantum field.

• Could you provide more details for the third part? Sep 12, 2021 at 8:49
• @Mauricio I think the punchline on Entanglement is that entanglement works essentially the same way mathematically in algebraic quantum field theories as it does in standard finite-dimensional quantum mechanics. Witten's notes (linked in the answer) are a wonderful exposition on that subject. Sep 12, 2021 at 15:14
• @Condo I will surely work it out at some point, but if it offers no new insight why even bother? When I asked this I was expecting a slightly larger comment on either how to do this relativistic description or a summarized enumeration of the main insights that it provides. Sep 12, 2021 at 15:40

When discussing relativistic quantum mechanics, the single particle picture is dropped completely

Not completely. You can still consider 0-particle and 1-particle states. The 0-particle state is usually called the vacuum, and can be denoted by a ket like $$|\Psi^{(0)}\rangle$$, which is annihilated by all the particle annihilation operators: $$c_i |\Psi^{(0)}\rangle = 0\;.$$

1-particle states look like: $$|\psi^{(1)}_i> = c^\dagger_i|\Psi_0\rangle$$

2-particle states look like: $$|\psi^{(2)}_{ij}> = c^\dagger_j c^\dagger_i|\Psi_0\rangle\;,$$ where the nice thing about the QFT formulation is that the bosonic or fermionic nature (the wave function symmetry) is explicit via the operator relations.

which makes the description of just two entangled particles difficult.

No more difficult than anything else really. The difficulty probably depends on how familiar with QFT one is.

However can some description be built from Dirac's equation or quantum field theory?

Yes, some description can be built. It is probably not helpful, though, in practice, since quantum computers are non-relativistic condensed matter systems. A familiarity with the Dirac equation can be helpful if the quantum computer is built, say, from Majorona fermions that emerge out of some types of condensed matter band structure.

• I think I get that second quantization might be helpful but beyond that, would identifying some relativistic symmetries or some relativistic equations allow to understand what is happening, for example with locality during a Bell test. Feb 22 at 19:31
• I'm not entirely sure. Relativistic quantum field theories are typically designed to have the causal aspects of relativity "built in." This means that we can avoid the problems with the single particle propagator $\langle \vec x|e^{-iHt}|\vec y\rangle$ allowing for propagation outside the light cone, but it forces us to consider many particles at once (in particular, anti-particles). This is more about "causality" than "locality." Locality in QFT is usual described by saying that the fields in the Lagrange density are evaluated at the same point in space.
– hft
Feb 22 at 20:01
• E.g., Local: $L \sim \int dx \phi(x)\phi(x)$. E.g., Non-Local: $L = \int dxdy\phi(x)\chi(x,y)\phi(y)$.
– hft
Feb 22 at 20:04