Is Probability conserved in RQM?

[LarryS]

Is probability (or probability current) conserved in RQM/QFT? I have never seen a simple answer to this question.

As always, thanks in advance.

 

[stevendaryl]

I think it's usually considered to be a sign of an inconsistency in a theory if probabilities don't add up to one at all times. Or it's a sign that the quantity that you thought of as probability isn't really a probability (which was the case with the current derived from the Klein-Gordon equation)

 

[DEvens]

Is probability of what conserved?

QFT conserves lots of things. Momentum, energy, angular momentum, various charges.

Other things are not so straightforward.

https://en.wikipedia.org/wiki/Lepton_number
https://en.wikipedia.org/wiki/Weak_hypercharge

Some things simply are not conserved. You can pretty much produce as many photons as you are prepared to expend the effort to produce. So the probability of observing a photon is not conserved.

 

[LarryS]

Is probability of what conserved?

QFT conserves lots of things. Momentum, energy, angular momentum, various charges.

Other things are not so straightforward.

https://en.wikipedia.org/wiki/Lepton_number
https://en.wikipedia.org/wiki/Weak_hypercharge

Some things simply are not conserved. You can pretty much produce as many photons as you are prepared to expend the effort to produce. So the probability of observing a photon is not conserved.

I should have said "Is the position probability conserved?".

 

[stevendaryl]

I should have said "Is the position probability conserved?".

In quantum field theory, there is not a definite number of particles. And particles of the same type are indistinguishable. So in QFT you would not ask "What is the probability that this particle is at location $x$?" Instead, you would ask: What is the probability of finding a particle at location $x$, with the understanding that finding a particle at one location doesn't preclude the possibility of finding an identical particle at another location at the same time.

 

[bhobba]

I think it's usually considered to be a sign of an inconsistency in a theory if probabilities don't add up to one at all times.

The problems with probability in relativistic QM were all resolved when going to Quantum Field Theory. Doing so also showed how they could be resolved in things like the Klein Gordon equation. Since QFT requires antiparticles negative probabilities in a current density are positive probabilities of the antiparticle.

Thanks
Bill

 

[Demystifier]

I should have said "Is the position probability conserved?".

There is no simple answer because there is no simple answer to an even more elementary question: What is the position operator for RQM/QFT?

 

[DEvens]

I should have said "Is the position probability conserved?".

Um... No it's not. The probability of finding a particle at a position isn't conserved. Why would you ever think it was? Particles move around.

I think you are struggling to ask a different question. Maybe you could ask what it is you are really trying to ask.

 

[LarryS]

There is no simple answer because there is no simple answer to an even more elementary question: What is the position operator for RQM/QFT?

Obviously, since it includes SR, QFT/RQM is considered physically more accurate than the original non-relativistic QM. So does that mean that the Born Rule is a nice approximation of reality but not fundamentally true?

 

[DEvens]

Obviously, since it includes SR, QFT/RQM is considered physically more accurate than the original non-relativistic QM. So does that mean that the Born Rule is a nice approximation of reality but not fundamentally true?

The first sentence has nothing to do with the second sentence. So it clearly does not mean that. Whatever you may mean by "fundamentally true" as opposed to true.

https://en.wikipedia.org/wiki/Born_rule

Depending on how you look at it, the Born rule is either a postulate of, or a derivable consequence of the rest of, quantum mechanics.

Again, it seems like you are struggling to ask some other question. Could you please ask that other question?

 

[bhobba]

Depending on how you look at it, the Born rule is either a postulate of, or a derivable consequence of the rest of, quantum mechanics.

Just on that you might like to look into Gleasons Theorem.

Thanks
Bill

 

[stevendaryl]

Obviously, since it includes SR, QFT/RQM is considered physically more accurate than the original non-relativistic QM. So does that mean that the Born Rule is a nice approximation of reality but not fundamentally true?

The Born rule works as well in QFT as in QM. In both cases, we have states, and we use the states to compute probabilities for measurement results, using the Born rule. The difference is that in single-particle, nonrelativistic QM, the "state" is described by a wave function giving a probability amplitude for finding that particle at a particular position, while in QFT, the "state" describes an indefinite number of particles.

 

[LarryS]

The first sentence has nothing to do with the second sentence. So it clearly does not mean that. Whatever you may mean by "fundamentally true" as opposed to true.

https://en.wikipedia.org/wiki/Born_rule

Depending on how you look at it, the Born rule is either a postulate of, or a derivable consequence of the rest of, quantum mechanics.

Again, it seems like you are struggling to ask some other question. Could you please ask that other question?

I am referring to the "conservation law for probability in quantum mechanics" as referred to in the link:
https://en.wikipedia.org/wiki/Probability_current#Continuity_equation_for_quantum_mechanics

 

[DEvens]

I am referring to the "conservation law for probability in quantum mechanics" as referred to in the link:
https://en.wikipedia.org/wiki/Probability_current#Continuity_equation_for_quantum_mechanics

Oh. Then yes, it is conserved in relativistic quantum field theory in exactly the same way.