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https://en.wikipedia.org/wiki/Spacetime_algebra#Relativistic_quantum_mechanics claims that the wave-function can be written using geometric algebra as follows:

$$ \psi=R(\rho e^{i\beta})^{1/2} $$

Furthermore, it is claimed that that probability measure is obtained as follows:

$$ \psi\tilde{\psi}=R(\rho e^{i\beta})^{1/2} \tilde{R}(\rho e^{-i \beta})^{1/2}=\rho $$

I am confused as to how one would write the normalization condition for this probability measure. I see two possibilities:

$$ \iiiint\psi\tilde{\psi}dxdydzdt=1 \tag{1} $$

$$ \iiint \psi\tilde{\psi}dxdydz=1 \tag{2} $$

The problem with 1) is that it seems to suggest that the event should happen once in space-time, as opposed to describing a wavefunction that remains normalized as it evolves in space-time.

The problem with 2) is that it seems to artificially select one parameter to be 'special' whereas in relativity my guts tell me it should be avoided.

Finally, this paper https://arxiv.org/abs/1102.2083 seems to suggest that the answer is 1). On the very first sentence of page 3 and it states:

[...] the normalization condition $\int \psi \tilde{\psi} d\tau=1$, where $d\tau$ is [the] volume of four dimensional space.

Is this correct; are we to normalize in space-time rather than space? This would mean that the probability of an event in space, after some time, would be zero. As an example, supposed I want to describe the probability of being born at a point in space. I know I was born at some location at some time. So the probability of me being born in space-time is one. Yet, the probability of being born at any time beyond my birth date is zero. Is this conceptually simplified example the class of events a space-time wave-function encodes?

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  • $\begingroup$ Is it just a question for a general Dirac field? I don't see any specific relationship of this question with geometric algebra. $\endgroup$
    – MadMax
    Feb 18 at 23:10
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What you've uncovered is motivation for the statement, "relativistic quantum mechanics is intrinsically a many-body theory". Standard model operators like creation and annihilation do precisely what you describe - have a particular particle's wavefunction cease to exist after a certain time.

Spacetime normalisation is the only Lorentz-invariant way to do it - it would not be physically sound for your wavefunction's normalisation to change between inertial frames.

Mass-energy conservation is written into the structure of a Lagrangian describing a theory (specifically, its time translation symmetry), not the normalisation convention.

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  • $\begingroup$ Interesting observation with the annihilation/creation operator - thanks! So taking a wave-function that starts to exists at the creation operator and ceases to exists at the annihilation operator: is it the space-time volume (four-volume) that is normalized to one, or is it the time-slices of the space volume (three-volume) that are each normalized to one? Are you saying that it is the integral over the space-time volume of the wave-function that must be normalized, because otherwise it would violate Lorentz invariance? $\endgroup$
    – Anon21
    Feb 19 at 0:49
  • $\begingroup$ Wow that is pretty clever. $\endgroup$
    – Anon21
    Feb 19 at 0:56

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