The uncertainty principle and moving reference frames

[Daniel Gallimore]

To what extent is the Heisenburg Uncertainty principle a statement about moving frames of reference? The ill-defined position of a particle seems to imply that one can never find an inertial frame of reference in which the velocity of a given particle is constant.

 

[PeterDonis]

To what extent is the Heisenburg Uncertainty principle a statement about moving frames of reference?

None. See below.

The ill-defined position of a particle seems to imply that one can never find an inertial frame of reference in which the velocity of a given particle is constant.

The usual position-velocity form of the uncertainty principle assumes non-relativistic QM, in which the rule about the laws of physics looking the same in different inertial frames doesn't really apply. If you really want to combine quantum mechanics with the principle of relativity, you need to look at quantum field theory, in which the formulation of the uncertainty principle requires more care.

 

[dextercioby]

@PeterDonis Well, the principle of relativity (Galilean or Lorentzian) is compatible with the theory of Quantum Mechanics for a finite number of degrees of freedom - as per the Heisenberg/Copenhagen microscopic/macroscopic cut.
Here's the argument by Fonda & Ghirardi. I buy it.

 

[PeterDonis]

the principle of relativity (Galilean or Lorentzian) is compatible with the theory of Quantum Mechanics for a finite number of degrees of freedom

That's not quite what I read the quote you give as saying. I read it as saying that the principle of relativity is compatible with the classical limit of QM. But in the classical limit we ignore the uncertainty principle since its effects are negligible compared to the size of the effects we are observing; so we can treat position and velocity as both having determinate values for purposes of the principle of relativity.

 

[dextercioby]

Drop the quote, then. What I want to say is that the so-called uncertainty relations typical for QM are unrelated to a possible failure of the principle of relativity in quantum mechanics. And switching from Galilean QM to Lorentzian QM (in the form of a QFT) will not "cure" this possible failure which you claimed to be true only in a non-specially relativistic QM.

 

[PeterDonis]

What I want to say is that the so-called uncertainty relations typical for QM are unrelated to a possible failure of the principle of relativity in quantum mechanics.

I'm not sure that's true; if we are not talking about the classical limit, then it seems to me that in order to define quantum states and operators so that you can formulate the uncertainty relations, you need to pick a particular frame of reference. In other words, I'm not sure that the usual non-relativistic QM equations are Galilean invariant. (In particular cases they might but, but I'm not sure they are in general.)

switching from Galilean QM to Lorentzian QM (in the form of a QFT) will not "cure" this possible failure

Why not?

 

[dextercioby]

The uncertainty relations are a consequence of non-commutativity of observables (equal time commutators). Upon a change of Galilean/Lorentzian reference frames, the observables are covariant under the corresponding symmetry group. That entails that the commutator is also covariant, therefore there is no conflict with the principle of relativity.

 

[strangerep]

As pointed out by Gerry Kaiser years ago, the non-relativistic (centrally-extended) CCRs (from which the non-rel uncertainty principle arises) are actually a consequence of the Poincare commutator between boost and momentum (spatial translation) operators. This can be seen by performing a contraction from Poincare to Galilei while remaining in a Poincare irrep.

(I can probably dig out the details if anyone wants them.)

 

[f95toli]

The uncertainty relations are a consequence of non-commutativity of observables (equal time commutators).

Indeed, whenever these questions come up it is useful to remember that the HUP applies to all pairs of non commuting variables; not just "regular" position-momentum.
A good example (since it is what I work on) would be that the HUP also applies to charge-phase in an electronic circuit (in the simplest case a regular LC circuit), i.e. the HUP applies even though there is nothing moving and there are no spatial coordinated involved.

 

[PeterDonis]

Upon a change of Galilean/Lorentzian reference frames, the observables are covariant under the corresponding symmetry group. That entails that the commutator is also covariant, therefore there is no conflict with the principle of relativity.

Ah, ok, that makes sense.

 

[vanhees71]

The usual position-velocity form of the uncertainty principle assumes non-relativistic QM, in which the rule about the laws of physics looking the same in different inertial frames doesn't really apply. If you really want to combine quantum mechanics with the principle of relativity, you need to look at quantum field theory, in which the formulation of the uncertainty principle requires more care.

Non-relativistic quantum theory is, of course, by construction Galilei invariant, i.e., the observable algebra is constructable by assuming that this algebra allows for unitary ray representations of the Galilei group (semidirect product of temporal and spatial translations, rotations, and boosts).

As turns out from this analysis, the Galilei group can be extended to what I'd call "quantum Galilei group", which is a central extension of the covering group of the classical Galilei group with mass as a non-trivial central charge.

The position-momentum (not velocity!) uncertainty relation follows directly from the definition of momentum as generators of translations, i.e., the (Heisenberg) subalgebra of the corresponding Lie algebra,
$$[\hat{x}_j,\hat{p}_k]=\mathrm{i} \hbar \delta_{jk}.$$
Of course, the position observable has to be derived from the generators of the boosts, but this is pretty straight-forward.

In the relativistic case, it's more subtle. There the construction of position observables is not so straight-forward. It turns out that there exists proper position operators for massive particles (note that in contradistinction from the Galilei group in the case of the Poincare group mass (squared) is a Casimir operator rather than a central charge) and massless particles with spin $\leq 1/2$. For these cases, of course, the Heisenberg uncertainty relation (for position and momentum, not velocity!) holds. A famous analysis of this uncertainty relation by Peierls and Landau [1] taking into account the finiteness of the speed of light shows that a wave-mechanical interpretation as in the non-relativistic case is inconsistent, and one has to take into account the possibility of creation and annihilation processes in scattering processes at relativistic energies. Massless particles with spin $\geq 1$ do not allow for the definition of a position operator.

[1] L. Landau, R. Peierls, Z. Phys. 69, 56 (1931)

 

[edguy99]

Indeed, whenever these questions come up it is useful to remember that the HUP applies to all pairs of non commuting variables; not just "regular" position-momentum.
A good example (since it is what I work on) would be that the HUP also applies to charge-phase in an electronic circuit (in the simplest case a regular LC circuit), i.e. the HUP applies even though there is nothing moving and there are no spatial coordinated involved.

Did not know this. Is there an example that would highlight the effect?

 

[f95toli]

Did not know this. Is there an example that would highlight the effect?

I can't think of a good reference that discusses this explicitly right now. MH Devoret's lecture "http://www.copilot.caltech.edu/documents/260-les_houches_devoret_quantum_fluctuations_electrical_circuits_1997.pdf" (the PDF can be found online) is still one of best references for quantum electrical circuits; but note that it was written in 1997, two years before the first superconducting qubit was demonstrated so whereas the theory (which as far as I remember does discuss the HUP for charge-phase) is fine it is a bit out of date.

Superconducting qubits are otherwise the most obvious demonstrations of this. These qubits are electrical circuits (capacitors, inductors and non-linear elements based on Josephson junctions) which either use charge (zero or one electron/Cooper pair on an island) or phase/flux (in the simplest case direction of current flow in a loop). By choosing your parameters you can decide if you want charge or phase to be your good quantum number. Modern qubit design such as transmons, xmons etc are a bit more complicated but the basic physics is the same.