# How to mathematically formulate the Two Slit Experiment in Special Relativity?

How to mathematically formulate the Two Slit Experiment in a Lorentz invariant framework.

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As several of us have argued beneath your other question here:

Special relativity version of Feynman's "Space-Time Approach to Non-Relativistic Quantum Mechanics"

quantum mechanics combined with special relativity needs one to use quantum field theory (or a theory that is stronger than that).

If one wants to study relativistic particle in the presence of slits, one may imitate it simply by putting various (absorbing) boundary conditions (for the fields whose excitations are the interfering particles) at the places where the boundaries of the material reside. The probability waves in space propagate according to the standard free equations, so interference measures the free particles' propagators.

In reality, the only relativistic particles whose interference may be observed in practice at present are photons. The mathematics of interference of individual photons is equivalent to the interference of classical electromagnetic waves - as they studied it since the early 19th century. The probability density is mapped to the energy density of an electromagnetic wave and the $E,B$ fields may be interpreted as the photon's wave function. The results for the interference patterns are, up to a normalization that can be determined from the total number of particles, identical to the case of the classical electromagnetic waves.

So papers for photons exist and the first ones were written in the early 19th century. Papers for other particle species don't exist because

1. they're simple sums of the well-known functions that govern the propagator of the corresponding probability waves in the space; for example, one only needs the simple, free one-particle Dirac equation to calculate the propagation of the electrons at any speeds

2. the interference can only be measured for photons whose mathematical description is a very old story. For example, relativistic electrons must have a wavelength comparable to the Compton wavelength of the electron, something like $10^{-12}$ meters, or even shorter (ultrarelativistic electrons). The corresponding inteference pattern would probably be too tiny to be seen

Cheers LM

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Thanks for the answer! I'll leave it open for a while so others can pitch in too. –  mtrencseni Jan 22 '11 at 8:44