What is it about?

In quantum mechanics the electron and other massive elementary particles are described by the wave function which is a function of commuting particle coordinates (x,y,z) or, equivalently, by the commuting momenta (p_x, p_y, p_z). Massless particles, like photons and gravitons, are also described by these variables, but, as it was discovered by Schwinger, now their coordinates are not commuting: [x,y] = i hbar p_z/ p^3, and therefore their wave functions are naturally described in terms of commuting momenta (p_x, p_y, p_z). We have shown that this leads to the quantisation of the massless particle helicity (spin) and duality between the helicity and the Dirac quantisation conditions.

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Why is it important?

Photons are particles that are surrounding us all the time. Their properties are important for understanding of our very large Universe and its evolution and are also important for our understanding of the behaviour of a single photon qbit that can be used in a future quantum computers.

Perspectives

While investigating the irreducible representations of the high-spin extension of the Poincare algebra, which was introduced earlier in our publication in JMP, I was studying the work of Schwinger and found his almost unknown results about massless representation of the Poincare algebra and non-commutativity of the corresponding coordinates. The communication with my colleagues working in non-commutative field theory revealed that this work of Schwinger remained unnoticed, and it was my intention to make it known to a larger auditory.

George Savvidy
National Center for Scientific Research "Demokritos" , Institute of Nuclear and Particle Physics

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This page is a summary of: Schwinger’s non-commutative coordinates and duality between helicity and Dirac quantization conditions, Journal of Mathematical Physics, August 2025, American Institute of Physics,
DOI: 10.1063/5.0279941.
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