Consistency of Relativistic QM

[bhobba]

Hi All

While going through my book on QFT for mathematicians (What Is a Quantum Field Theory?
by Michel Talagrand) I came across the following:

'Quantum Mechanics, in its standard formulation, is not compatible with Special Relativity,
and it is very difficult to reconcile these two theories. Early attempts in this direction go
under the name of “Relativistic Quantum Mechanics”. This theory, which is the subject of
numerous textbooks, runs into severe inconsistencies, and we will not try to describe it.'

I knew there were problems (e.g., negative energy in the Dirac equation). Also, QFT is much more elegant, IMHO, as it treats fields and particles on the same footing. But actual inconsistencies?

Can people shed some light on this?

Thsanks
Bill

 

[renormalize]

Can people shed some light on this?

Here's a recent thread on that topic:

Graduate Thread 'How to fix Relativistic QM so it's consistent?'

Oct 25, 2025

What attempts have been to resurrect RQM (in order to ditch the fields' notion)?
Besides the appearance of negative energies (which I guess it's a blasphemy in physics), what other issues are there in RQM? and do they reappear in QFT?

• mad mathematician
• Replies: 23
• Forum: Beyond the Standard Models

 

[gentzen]

You know https://arxiv.org/abs/quant-ph/0608140 (1951 Lectures on Advanced Quantum Mechanics Second Edition by Freeman J. Dyson)?

In this course I will go through the one-particle theories first in detail. Then I will talk about their breaking down. At that point I will make a fresh start and discuss how one can make a relativistic quantum theory in general, using the new methods of Feynman and Schwinger. From this we shall be led to the many-particle theories. I will talk about the general features of these theories. Then I will take the special example of quantum electrodynamics and get as far as I can with it before the end of the course.

This is an historic equation, the Klein-Gordon equation. Schrödinger already in 1926 tried to make a RQ
theory out of it. But he failed, and many other people too, until Pauli and Weisskopf gave the many-particle
theory in 1934 [12]. Why?
Because in order to interpret the wave-function as a probability we must have a continuity equation.
This can only be got out of the wave-equation if we take j as before, and
ρ =i~2mc2ψ∗ ∂ψ∂t −∂ψ∗∂t ψ (7)
But now since the equation is 2nd order, ψ and ∂ψ
∂t are arbitrary. Hence ρ need not be positive. We have Negative Probabilities. This defeated all attempts to make a sensible one-particle theory.

The theory can be carried through quite easily, if we make ψ describe an assembly of particles of both positive and negative charge, and ρ is the net charge density at any point. This is what Pauli and Weisskopf did, and the theory you get is correct for π-mesons, the mesons which are made in the synchrotron downstairs. I will talk about it later.

All negative-energy states are normally filled by one electron each. Because of the exclusion principle
transitions of ordinary electrons to these states are forbidden. If sometimes a negative energy state of
momentum −p energy −E is empty, this appears as a particle of momentum p energy +E, and the opposite
charge to an electron, i.e. an ordinary positron.
Thus we are led at once to a many-particle theory in order to get sensible results. With spin-0 particles,
to get positive probabilities. With spin-1/2 particles, to get positive energies.
The Dirac theory in its one-particle form cannot describe properly the interaction between several
particles. But so long as we are talking only about free particles, we can describe them with one-particle
wave-functions.

But of course, even if you don't know this text, you certainly know these arguments. But maybe the last bold sentence above (bold by me) helps you at least a bit to understand what people mean by "it is very difficult to reconcile these two theories": Using the non-relativistic Schrödinger equation, describing the interaction between several particles is no real problem. So it must be compatibility with Special Relativity which is responsible for those troubles.