# How do I normalize the relativistic wavefunction in general?

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?

• Is it just a question for a general Dirac field? I don't see any specific relationship of this question with geometric algebra. Feb 18, 2021 at 23:10