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In the context of the second quantization and the use of fields in the canonical quantization, the canonical momentum of the field is defined as the derivative of the field by the time coordinate. But if we're talking relativistically, shouldn't it be the derivative of the field by the proper time? What am I missing?

Thanks

Rafael Gomes
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1 Answers1

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The second quantisation you mentioned is an equal time quantisation so it is specific to the frame one starts with, and for this reason time and spatial indices are not treated equally. For details, one can see ,for example, the canonical quantisation of scalar field from Srednicki's QFT Chapter 3. However, we do need to check that the canonical quantisation is compatible with Lorentzian transformations, and one way to do this is to check the following diagram commutes: (for notational convenience I only write one particle state. But this should be checked for $n$ particle states )

$$\require{AMScd} \begin{CD} p^u @>{Quantization }>> |p^u >;\\ @V{\Lambda}VV @V{U(\Lambda)}VV \\ \Lambda^u{}_vp^v @>{Quantization}>>|\Lambda^u{}_v p^v > ; \end{CD}$$

where $U(\Lambda)$ is according to Srednicki's notation the representation of Lorentzian transformation.

user110373
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