Is there a four dimensional form of Born's Rule -redub

Question

Generalizing Born's Rule for 4-dimensions $x_4$, write

$$\langle a\rangle = \int\Psi A\Psi^* \mathrm{d}x_4$$

1. Is this consistent with quantum mechanics?
2. Is this a generalized form of the Born's Rule for the phase space of the wave function?

@DavidZas I was thinking of a four dimensional space where Time is not a parameter. There been recent work on Time Reversed light pulses.

http://prl.aps.org/abstract/PRL/v106/i19/e193902 and the PHYSORG description http://www.physorg.com/news/2011-05-physicists-time-reversed-pulses.html

And this leads me to wonder about abstract phase space, and I mean phase both in the epoch angle of a wave function and the phase space of a Hamiltonian (which yes, is very confusing). The way I see it, the wave function implictly has the Action of phase space built into it.

$\psi = e^{i(\omega t - k x)} = e^{\frac{i}{\hbar }(\mathbb{E} t - \mathbf{p} x)}$

Where the phase and the action are connected by

$\frac{\mathbb{E} t - \mathbf{p} x}{\hbar }$

Imagine dividing the phases by the action of the particle It's a bit like there's a blob (note the very precise language) of phase space wobbling about like the liquid drop model of the nuclear physics. Only this is a more abstract phase space that's oscillating.

So, again, would Born's Rule apply to such a system?

Actually after rereading what I just written, I'll be surprised if anyone replies this time! :)

Answer 1

Of course this is consistent -- the calculation of an expectation value is defined in fact for operators in a totally generic Hilbert space. If you want that Hilbert space to describe 4 spatial dimensions, more power to you.

On the other hand, if you are imagining $x_4$ as a time dimension, it's important to remember that Q.M. treats space and time asymmetrically. (Or more concretely, time is a parameter and not an operator, which is really just to say that there's nothing special at all about time averaging in Q.M.!)

Answer 2

Your starting point is not a "generalization of the Born rule". Assuming that you properly defined a Hilbert space that contains your $\Psi$ as a vector, your starting point is an instance of the Born rule.

The Born rule takes a vector $V$, in a Hilbert space, $H$, on which there is an inner product $(\cdot,\cdot)$, by definition, and tells us that the expected value of an observable represented by the operator $\hat A$, which acts on the vector $V$ in that Hilbert space, is $\bigl<\hat A\bigr>=(V,\hat A V)$. There's no mention, as you see, of 3-dimensional space. I suspect you're thinking in terms of the Schrodinger equation, where the Hilbert space is the infinite-dimensional Hilbert space of complex-valued functions on $\mathbb{R}^3$, $L^2(\mathbb{R}^3)$, and seeking to generalize to space-time by introducing the Hilbert space $L^2(\mathbb{R}^4)$. Since this is just another Hilbert space, the Born rule applies to it.

One can do this as Mathematics, but to my knowledge this construction is not used in Physics. Instead, the usual route is to introduce operator-valued distributions as part of quantum field theory. The approach that I have suggested you are proposing could be seen as the one-particle subspace of the many-particle Hilbert space of a quantum field theory (I usually try to avoid all talk of particles, but I think putting this statement in field-theoretic terms wouldn't be helpful here). If you want to move to Minkowski space, I believe you will not be able to avoid moving to quantum field theory. Or, at least, that you will have to be able to present very explicitly what the relationship is between quantum field theory and whatever you move to.