How can a particle's position be random and uncertain in quantum mechanics if it is already pre-determined in relativity?

Question

In relativity, to my knowledge, the path of an object is described by its worldline in spacetime, and since time is a part of the spacetime geometry, an object's worldline--in a sense--always exists on this “block of time” as I heard the PBS spacetime say. But in quantum mechanics (the Copenhagen interpretation), I’ve heard that a particle will exist in (superposition of being in) multiple places at once until it is observed and the wave function collapses and it has a single position.

So my question is, how could these two viewpoints be reconciled? Apologies if I got anything wrong.

Answer 1

There are two problems here, one on the quantum mechanical side and one on the relativistic side.

Interpreting quantum mechanics

First, you seem to be imagining that quantum particles "really" are classical, with well-defined trajectories which we simply can't measure because of the uncertainty principle. That is, you treat quantum mechanics as just classical mechanics viewed through a smudged lens. That is not a good way of thinking about it: nature is much weirder than that. (I try to give a better explanation here.)

To avoid confusion, you should refrain from using phrases such as "the actual position of the particle" or "the path the particle took". Imagine trying to explain to a blind person how a screen's color is fading from white to black, and them asking "okay, but was it actually black or white right in the middle?" It's just not a valid question; there is no answer.

In the modern formulation of relativistic quantum field theory, we define quantum fields on all of spacetime $\phi(t, \mathbf{x})$. So the quantum field can be defined for all times from the get go, but this does not mean that it represents a definite number of particles doing definite trajectories, any more than a screen having a color at any time means it's always either black or white.

Interpreting relativity

The second problem is with the interpretation of relativity. I think you're alluding to Putnam's block universe argument. The argument is essentially that, since things that will occur in the future in my frame have already occurred in somebody else's frame, due to relativity of simultaneity, the future must "already" "exist", so it must be predetermined. However, one shouldn't confuse the mathematical formalism of a theory, i.e. the easiest way to set it up, with its ontology, i.e. what it states about reality.

Some summarize this by saying "the map is not the territory". If you have a road map with a grid of latitude and longitude lines, that does not mean the real ground is covered in giant lines. The lines were just drawn to make the map more useful. Not everything on the map reflects reality.

Similarly, relativity pushes us to set up calculations so that everything is already defined for all times, but this is not necessary. For example, in the ADM/3+1 formalism of general relativity, things are specified only at a single time, then propagated forward in time. So in this map, the future does not exist, only the present. This is essential for numerical simulations, because how would you have a computer compute the future if it had to know it already?

The point is that there are multiple ways to set up relativity, and all of them have different features. Since they all make the same concrete predictions, science can't choose one. (This is why I get annoyed at grand statements about how relativity tells us what spacetime really is, when it's really just a feature of the one map the speaker has used.)

If you insist on a particular interpretation of relativity (the block universe) and a particular interpretation of quantum mechanics (Copenhagen), then there is indeed a contradiction, because the Copenhagen interpretation requires an indeterminate future. But that doesn't mean the underlying theories contradict each other, it only means that these two particular ways of talking about them don't mesh; you'll need to swap out one or the other. I apologize for not making any strong statements here, but this agnosticism is the only scientifically tenable position.

Answer 2

The answer to this is that relativity is actually a theory of space-time geometry, not a theory of matter in motion. You can insert whatever kind of moving matter you want on top of it, but the essence of special relativity is simply that space-time transforms under the Lorentz transformations (or generally the Poincare group), with the Minkowski distance the appropriate notion of distance between events. It's essentially a theory of the "background" on which your theory of motion exists, not the theory of motion itself. The "theory of motion" found in "special relativity" that has definite positions, etc. is really a suitably-modified Newtonian mechanics and not strictly SR proper. But you can likewise add a suitably-modified quantum mechanics on top as well. You could even add in totally fictitious kinds of physics as in an imaginary universe - the possibilities are endless. The point is that none of them are "SR", but rather added theories of motion, and that SR is the common thing shared by all these universes.

The coupling between the background and the dynamic theory comes from the assumption of the principle of relativity, that is, that the dynamic laws, whatever they are, should continue to work even after transforming the system by arbitrary symmetry transformations from the Poincare group.

Answer 3

Background

In classical physics (relativistic or not), the position of particle $\mathbf{x}(t)$ and momentum $\mathbf{p}(t)$ are deterministic. What I mean is that given $\mathbf{x}(t_{0})$ and $\mathbf{p}(t_{0})$, $\mathbf{x}(t)$ and $\mathbf{p}(t)$ for $t > t_{0}$ can be determined if you know the dynamics of the system (i.e. Hamiltonian $H$).

In quantum mechanics (QM), observables like position and momentum cannot be precisely known simultaneously (not because of numerical error or experimental error, it is the property of nature itself). Mathematically, it is because the position operator $\hat{\mathbf{x}}$ and momentum operator $\hat{\mathbf{p}}$ are not commute (i.e. $[\hat{\mathbf{x}}, \, \hat{\mathbf{p}}] = \hat{\mathbf{x}} \hat{\mathbf{p}} - \hat{\mathbf{p}} \hat{\mathbf{x}} \neq 0$, which is related to uncertainty principle).

But in QM, what is deterministic is the evolution of probability density. For example, the probability density of finding a particle at position $\mathbf{x}$ and at time $t$: $|\Psi(\mathbf{x}, t)|^{2}$ is governed by the following dynamics (Schrodinger Equation) \begin{align} -\frac{\hbar^{2}}{2m} \nabla^{2} \Psi \; + \; V(\mathbf{x})\Psi(\mathbf{x}, t) \; = \; i \hbar \frac{\partial \Psi}{\partial t} \end{align}

Your Question

The compatibility issue between QM and special relativity (SR) is not about the unpredictability of the precise position and momentum value, it is about whether the dynamics of QM system (e.g. Schrodinger Equation) is invariant (keeping the same mathematical form) or not under Lorentz transformation.

For Schrodinger Equation that I write above, it is not Lorentz invariant. In fact, Schrodinger equation is Galilean invariant (Newtonian regime, not relativistic regime). There were lots of effort in developing QM systems that are compatible with special relativigy (i.e. invariant under Lorentz transformation). Examples include, Klein Gordon Equation, Dirac Equation. Ultimately, a self consistent theory (if I am not mistaken :)), Quantum Field Theory (QFT) that is compatible with SR, is developed.

Answer 4

So when a particle's position is measured, we can say that the particle is really located in some region of space. As time progresses without measurements, the region of where it could be grows according to the uncertainty in its momentum. A first understanding of relativistic QM just makes sure that this region is bounded by the starting region's light cone. relativity is much more concerned with the idea that no information is proven to move faster than the speed of light, than with the idea that nothing is uncertain.

With that said, in the usual description of quantum mechanics there is an instantaneous information propagation which did trouble physicists for a long time, called entanglement. The full pursuit of this feature has convinced most physicists that reality cannot be described in the "local" vocabulary that special relativity would like to use. However we now appreciate that this will never propagate usable information faster than the speed of light: the information is hidden in a correlation between two systems living a great distance apart, and cannot be observed until both measurements are brought back together for comparison.

Answer 5

QM and SR are actually compatible, it's QM and GR that is not.

The uncertainty relationship in QM manifests itself in QFT as the uncertainty in the number of particles. ie particle conservation is lost, this is the origin of virtual particles which we see in Feynman diagrams.

Unlike QM, SR or even GR, QFT is not rigourously defined. Constructive QFT is one attempt to do just this - however they cannot show the existence of even one interacting QFT. They can in fact, show the existence of a free QFT rigorously.

This is rather like saying we can rigorously construct the number zero, but one, two, three and so on are constructed by hand-waving!

Answer 6

The viewpoint of special relativity is that under any Lorentz transformation (switching from one constant velocity frame to another), stuff should never go faster than light. The viewpoint of quantum mechanics, from my viewpoint, is that governing equations should be sought which lead to quantization of things like atomic energy levels to discrete values, as has been observed in spectroscopy. These can be reconciled by finding a governing equation which ensures all of the above: both never letting information travel faster than light under Lorentz transformations, and leading to quantization of particle energy levels in the manner observed in nature. The Klein-Gordon equation was the first attempt to make this work, but it didn't. Eventually it was found that the Dirac equation fulfills the desired properties. Freeman Dyson's book Advanced Quantum Mechanics gives background on the development of special relativistic quantum mechanics within the first ten pages.

My apologies if this isn't the most rigorous or professionally interpreted explanation.

Answer 7

Relativity is significant only velocities that are significant fractions of that of light. The uncertainty in time and space of quantum mechanics are inverse to energy and momentum. Therefore, relativity and quantum mechanics are not both important to same problem.