Spacetime and uncertainty principle

Question

I only have limited knowledge of relativity and quantum physics but as far as I know, the uncertainty principle relates the uncertainty of space and momentum of a particle. Einstein however, explained that space and time are tied together and the real fabric of the universe is spacetime through which all objects navigate.

It feels as if space uncertainty should therefore be spacetime uncertainty. Is this wrong? Can it be that you know the position of a particle but not the exact time when it was there, and that that gives rise to uncertainty in momentum? If so, wouldn't this be a more elegant way to express the uncertainty principle?

Googling for "spacetime uncertainty" gives papers that go far over my head. While my math is good, we barely touched on quantum mechanics in physics.

Answer 1

the uncertainty principle relates the uncertainty of space and momentum of a particle.

It is one of the basic foundation stones of quantum mechanics, tied up with the solutions of quantum mechanical equations. Quantum mechanics is a successful theory describing the behavior in the microcosm of particles.

The Heisenberg Uncertainty Principle , HUP, is described mathematically by the commutation relations of quantum mechanical operators operating on the solutions that describe the position and momentum of a particle. In general it concerns pairs of observables and there are a number of pairs that display a HUP uncertainty, not just momentum and position.

Einstein however, explained that space and time are tied together and the real fabric of the universe is spacetime through which all objects navigate.

You are describing General Relativity. This is a classical theory applying to large dimensions in space and time. It is not quantized. The quantization of gravity is an ongoing research subject.

The HUP relations will exist in the appropriate format of the quantized General Relativity observables, once there is agreement on the quantization of gravity.

It has no meaning to mix up the two systems otherwise.

Answer 2

No. The Uncertainty Principle has to do with the act of measuring. Basically, you cannot simultaneously measure both position and momentum to an arbitrary degree of accuracy. The more accurately you meausre one, the less accurate your measurement of the other becomes. The uncertainty in momentum , as far as I know, won't result from your not knowing when the particle was at a particular place.

Answer 3

The question has not been addressed fully so far. The question is better re-framed as, "Can the Uncertainty Principle be written in co-variant form?" and the answer is yes. For example I can consider the four vectors (x,y,z,ict) and (px,py,pz,iE/c) as conjugate and write the uncertainty relation between them.

Answer 4

I love your question, but you might be thinking of the uncertainty principle incorrectly. The proper form of the uncertainty principle is about the spread in values of independent measurements on an ensemble of similarly prepared particles. It does not necessarily have to do with the uncertainties of any individual particles (as many popular science treatments suggest).

So in a sense, yes, due to special relativity any length L will have a difference in time associated with it in the sense that a moving observer would observe synchronized clocks placed L apart to be out of sync by a well defined amount. And yes this length L could be the spread in observed positions of an ensemble of similarly prepared particles. And if you put clocks there they would be out of sync by an amount predicted by special relativity, but this does not necessarily mean that one of the individual particles that were measured had any uncertainty in time associated with them.

In fact, I think your question is a pretty good proof that particles are not in fact individually "spread out in space" otherwise they would indeed have to be "spread out in time". Different parts of the particle would have to be older than other parts of the particle when viewed from a moving frame (i.e. they have different proper times). But when we measure a particle we find a single unified particle that has aged a common amount (as could be determined by whether it decays or not via half life).

Similarly, it doesn't seem to make sense for the particle to be spread out in coordinate time either. Consider that there is an asymmetry in space and time in the sense that, if you measure a particle at a specific time over all of space you are bound to find it. But if you measure a particle at a particular point in space over all of time, you are not guaranteed to find it at that position. The particle may never be measured at that position, but it has to be able to be measured to be somewhere at all times, so it isn't really even clear what having a spread in time or uncertainty in time would really mean. If there was uncertainty if a particle existed at a certain time, it would have to be able to be not found anywhere at a specific time. But this is AFAIK experimentally refuted for free particles that are not undergoing annihilation or anything like that.

Answer 5

Can it be that you know the position of a particle but not the exact time when it was there, and that that gives rise to uncertainty in momentum? If so, wouldn't this be a more elegant way to express the uncertainty principle?

There is a relativistic version of the uncertainty principle called the canonical commutation relations that you can learn about in books in quantum field theory, such as "Quantum Field Theory for the Gifted Amateur" by Lancaster and Blundell (Chapter 11).

But you have a common misconception about quantum theory and the uncertainty principle that should be addressed before saying anything about relativistic quantum theory.

If you take quantum equations seriously as a description of how the world works, then a particle is never at a single well-defined position. Rather it is described by a wave function that can undergo quantum interference in a way that depends on what interactions are happening at each point, so you can't say the particle is in a single location. There are versions of the particle in multiple locations that can interfere with one another in interference experiments, see "The Fabric of Reality" by David Deutsch Chapter 2 for a popular account.

To get the probability of finding the particle in a region when you measure it you integrate the wavefunction over that region. But even this isn't quite right, because the measurement device is also governed by quantum theory. So what happens is that to be accurate you have to describe the measurement device and the particle in a quantum model. When you do that you find that when information is copied from the particle to the measurement device there is a set of states that act approximately like the objects you see around you in terms of having a position and momentum:

https://arxiv.org/abs/quant-ph/0306072

Their states are actually not at a single position and momentum but are highly peaked around a particular position and momentum in each of the states. And there isn't a single state but multiple states that evolve approximately autonomously and systems are correlated to form layers that each act a bit like the universe as described by classical physics:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/quant-ph/0104033

The uncertainty principle places lower limits on how a state is spread in position given how much its momentum is spread and vice versa. The particle isn't a single location or momentum it is a blob spread out in position and momentum and the uncertainty principle places limits on how concentrated the blob can be. The relativistic version of this is just a blob that respects Lorentz invariance and some other restrictions.