Gupta-Bleuler

In quantum field theory, the Gupta–Bleuler formalism is a way of quantizing the electromagnetic field. The formulation is due to theoretical physicists Suraj N. Gupta and Konrad Bleuler.

Firstly, consider a single photon. A basis of the one-photon vector space (it is explained why it is not a Hilbert space below) is given by the eigenstates |k,ε_μ〉 where k, the 4-momentum is null (k²=0) and the k₀ component, the energy, is positive and ε_μ is the unit polarization vector and the index μ ranges from 0 to 3. So, k is uniquely determined by the spatial momentum ${\displaystyle {\vec {k}}}$. Using the bra–ket notation, we equip this space with a sesquilinear form defined by

where the ${\displaystyle 2|{\vec {k}}_{a}|}$ factor is to implement Lorentz covariance. The metric signature used here is +−−−. However, this sesquilinear form gives positive norms for spatial polarizations but negative norms for time-like polarizations. Negative probabilities are unphysical, not to mention a physical photon only has two transverse polarizations, not four.

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