# Quantum mechanics and relativity

1. May 23, 2007

Has Quantum mechanics been unified with relativity (general and special)? Is that the objective of quantum loop gravity? and if unification of the two hasn't occurred, how close are they to getting it? How much progress has been made?

2. May 23, 2007

### robphy

You might find interesting discussion at
"Beyond the Standard Model"
https://www.physicsforums.com/forumdisplay.php?f=66

3. May 23, 2007

### MeJennifer

No.

A book I highly recommend is "The Trouble With Physics: The Rise of String Theory, the Fall of a Science, and What Comes Next " by Leo Smolin.

Last edited: May 23, 2007
4. May 25, 2007

### MathematicalPhysicist

well, it's not accurate.
SR has been meshed with QM, in the form of QCD and QED, also known as QFTs.

5. May 25, 2007

### Demystifier

Even this is not without problems. One problem is the lack of a particle position operator. See e.g. the introductory paragraphs in
http://xxx.lanl.gov/abs/0705.3542

6. May 25, 2007

IMO, the lack of a position operator is secretely equivalent to an incorrect assumption in QFT.

In relativity we learn that position is what a rod measures. The lack of a position operator thus amounts to an assumption about our measuring rods: that they are heavy and thus classical, because the outcome of a classical experiment is a c-number rather than a self-adjoint operator.

One can formulate an analogous argument about time: time is what a clock measures, but Pauli's theorem asserts that there is no self-adjoint time operator, lest the Hamiltonian be unbounded from below. This is clearly a paradox, since the outcome of a physical clock-reading experiment must be an observable. The resolution is again that we implicitly assume our clock to be heavy and thus classical.

In a typical experiment, this assumption of heavy rods and clocks is an excellent approximation; it just means that the state of our measuring device is not affected when we observe it. However, it becomes a serious problem in quantum gravity, because a heavy rod or clock will interact with gravity and collapse into a black hole.

This is specifically a gravity problem. For other interactions, we want our measuring devices to have small charge (so that the fields are not disturbed) and large inertial mass (so the device does not recoil as a result of the observation). This assumption is consistent except for gravity, where the charge is related to the heavy mass.

One may note that this argument only applies to QFT but not to QM, i.e. QFT in 0+1 dimensions. The problem is that a physical observer recoils when she makes an observation, but there is nowhere to recoil in zero space dimensions. But it is QFT which is interesting in QG since GR is a field theory.

From this perspective, it is obvious how QFT must be modified to be compatible with gravity: introduce an explicit quantum operator which describes the clock- and rod-carrying quantum observer. One then expects to recover QFT and GR when the observer's mass M -> infinity and M -> 0, respectively.

7. May 26, 2007

### Chronos

Which is a problem in QFT - it assumes a background. But once you inject a background, it works extremely well. It's very confusing. The bottom line, IMO, is neither GR or QM is fundamentally correct. I would bet GR is more nearly correct, but not quite perfect. It does break down at the smallest of scales, but is that truly a flaw? It works so dang well at every other scale it is impossible to ignore. I dearly love QFT and all the tinker toys, but GR rules the universe at large. Some form of GR must be correct, IMO.

Last edited: May 26, 2007
8. May 26, 2007

### jambaugh

Particle position is not an observable in classical relativistic mechanics. Why should it be in relativistic QM?

Let me point out that we do not measure positions of particles in the sense of a device which registers a coordinate. Rather we measure particle counts or particle fluxes in volumes or through areas which are small and centered at given positions. Spatial position like time is a label we put on our devices, not an internal property of the particle. This again occurs both classically and quantum mechanically in relativistic mechanics.

There are perfectly good particle number operators for regions of space (at given times) in relativistic QM and QFT. Using the Gupta-Bleuler technique you even get perfectly good positive definite probability interpretation.

(There's no need to invoke Bohmian "interpretations". ;-)

Regards,
James Baugh

9. May 26, 2007

### Anonym

Consider classical relativistic mechanics (no EM and gravitation fields present). The single material particle remains the same notion as originally introduced by I.Newton.

Regards, Dany.

10. May 26, 2007

### jambaugh

Yes. It is what we consider as properties of the particle which changes as we unify space and time. In non-relativistic mechanics position was an observable and time was not. Thus we say we e.g. measure the momentum and the position of the particle at a given time.

Now we can't treat time and space differently once we relativize (either Galilean or Einsteinian relativity b.t.w.) and unify space with time. Either you imagine a clock built into the object telling you what time it is or you must treat both space and time coordinates as parameters.

Hence in the more modern description you e.g. measure the existence of a particle with some momentum at a given space-time event. Time and space both are parameters you attach to the act of observation and not treat as properties of the object. Especially think about the fact that in unified space-time a dynamic object is a world-line. The object observed is its dynamic evolution. Solving the dynamic problem then is less about the evolution of the object as much as it is about the equivalence of differently expressed observables e.g. seeing a particle at a given space-time point with given momentum equates to seeing the particle at a later space-time point with a given momentum. Said equation being defined by the dynamics.

As you move more toward field theory then your observables become observations of distributions of stress-energy, spin, and gauge charge.
Space-time coordinates are parameters distinguishing different (though dynamically related) acts of observation.

This point seems to be lost on Nikolic,DeMystifier as well as many many other theoretical physicists. It is a subtle point but an important one.

One other point. You can recover spatial position information if you make certain assumptions about the physical object, e.g. that it is a point mass with no intrinsic spin. In Einsteinian relativity your observables are the momentum, angular momenta, and "boost angular momenta", the last you can think of as mass moments relative to the origin.

Note that angular momentum and mass moments are defined relative to a space-time origin and inertial frame. I understand this best by noting that each observable corresponds to a generator of the corresponding kinematic group, in this case the Poincare group of translations, rotations, and boosts. But when you select out a Lorentz subgroup of the Poincare group there is a choice of such subgroups corresponding to the choice of space-time point left invariant by the Lorentz transformations.

Then in the same way selecting out the rotation subgroup of the Lorentz group depends on a choice of time axis left invariant by the rotations. When you are done you can then define the coordinate position of a (spinless) point object as the spatial origin about which the mass moments and angular momenta are all zero.

Classical this requires you measure the linear momenta to get the mass and then measure all the moments and angular momenta. From these you can write down a function expressing the coordinate positions (again under specific assumptions about the nature of the particle). Quantum mechanically these do not all commute so you can't simultaneously measure them. At best you can calculate expectation values and define an average center of mass/rotation.

SideNote: Some refer to "quantum geometry" or "non-commutative geometry" as a generalization of analytic geometry wherein coordinate observables no longer commute with each other. What I describe is not this situation. But also it points out the retrograde nature of considering non-commutative geometry seriously in physics. In the relativistic setting coordinates are parameters and not observables so upsetting their commutativity is i.m.n.s.h.o silly.

Regards,
James Baugh

Last edited: May 26, 2007
11. May 27, 2007

### Anonym

Thank you for your explanations. However it seems to me that you make the opposite statements simultaneously.

Now I am seating in my office. And watch my clock.” Time and space both are parameters you attach to the act of observation and not treat as properties of the object.” So far I do not see any problem neither with position nor with time (your original statement that with position I have). Before I will consider the motion of another object and whether it is the external or internal information (his t,x,y,z), I ask how I will communicate the information to him. The information and the communication are entirely different notions. It does not a matter whether I tell to him or him telling to me. The problem is how to do that. To the best of my knowledge that problem originally was considered and solved (in sea navigation) by Galileo (synchronization of clocks).

You are moving too fast for me. I am still within v<<c and within classical physics. But I would like to be clear: my “inclination” is to lead you to the opposite conclusions in the relativistic QM.

I feel uncomfortable here (too much high IQ people around). I would like to have your and OP permission and agreement to ask Zz to move us home.

Regards, Dany.

P.S. The BM people deny ultimate validity of SR. I may discuss with them only sex.

P.P.S. “It does not a matter whether I tell to him or him telling to me”. Sorry, it is wrong, but let us restricts ourselves now with communication of the complete information.

Last edited: May 27, 2007
12. May 27, 2007

I strongly vote for the first alternative. The way to implement this mathematically is to expand all fields in Taylor series, and formulate everything in terms of Taylor data rather than field data. The point is that Taylor data contains information both about the fields themselves (the Taylor coefficients) and the clock-carrying observer's position (the expansion point).

But to know where and when this act of observation takes place, you must read off physical rods and clocks. Doing this will change the quantum state of the rods and clocks. This physical effect is ignored if you treat space and time as parameters.

This is not an idle point about interpretation, because it leads to different math: new gauge and diff anomalies which generalize the Virasoro algebra to higher dimensions. These don't arise in field theory proper because the relevant cocycles are functionals of the observer's trajectory.

To do canonical quantization in a manifestly covariant way (not quite successfully), I invented a quantization scheme where dynamics is treated as a constraint in the phase space of arbitrary histories. Instead of first solving the constraint, i.e. to coordinatize histories by positions and momenta at time t = 0, I prefer to quantize in the history phase space first and impose dynamics as a constraint a la BRST afterwards. This has the advantage that the constraint algebra of GR becomes the full spacetime diffeomorphism algebra rather than the messy Dirac algebra.

13. May 27, 2007

### jambaugh

Yes it seems so but the key point is that one is defining coordinates of the object as coordinates of those observational devices which register certain values for other observables. It is similar to electrical potential which is not an observable but which is defined as a relationship between objects which satisfy certain relative properties with regard to actual observables, namely charge and energy observables.
And the problem was reopened by Einstein who noted that you must look at the means of clock synchronization, namely exchanges of e-m pulses or other information carrying devices.

I like to think of these parametric quantities --position, time, angular orientation, phase-- in Lie group terms. The relative configuration coordinates of two isomorphic objects are the parameter values for the group necessary to bring either object into correspondence with the other, i.e. to transform them so their observables coincide. The actual observables are in one to one correspondence to generators of the transformation group.

The canonical formulation is simply a means to embed the physical transformation group, or rather its Lie algebra, within a specific class of Lie algebras defined by Poisson brackets on functions of phase space. Look at the expositions of constrained Hamiltonian systems to see how complicated this gets when you parse through the implications of which quantities are and which are not observables.

14. May 27, 2007

### jambaugh

Yes, you can do what you like on paper. But if I hand you an electron how do you actually observe its internal clock? Treating a quantity as an observable is not just a matter of choice or opinion. You must identify the laboratory procedure which implements the act of observation. There are some semantic conventions but that only gives a little wiggle room.
Not just ignored. They actually are not observed per se. You are defining space-time coordinates as the addressing system for the rods and clock-ticks. But you bring to mind an interesting issue. Recall the argument behind the Plank scale. My thesis advisor considered a more general problem of resolving field measurements over a range of positions to within a certain positional resolution and calculated the limit at which via Heisenberg uncertainty your measurement process must induce a gravitational collapse. It suggest a much larger scale than the Plank scale at which field theories loose validity.
Different math for certain, but what about the physics? There is an aspect of field theories in general which lead to the divergences and anomalies in the mathematics which in turn require special treatment. Said problems of course get worse when you attempt to quantize gravitational fields and this suggests that in spite of the great success of QFT+Standard Model, one should reconsider the format in which the physical theories are formulated. This e.g. is what string theory does however i.m.n.s.h.o. String theorists fail to actually look at the actual source of the failings of field theories and replicate the same problems. Those problems I assert are the invocation of ontological meaning to purely parametric entities, namely the space-time manifold and hence the strings and branes they quantize.
I'd like to see the details. However I feel that the whole canonical formulation is inherently flawed, said flaws undermine its generalization to the unification of gravitation with the other gauge forces.

I'm not as well versed on loop quantum grav. and on what its foundations are built. But I suspect the same pathologies exist there, namely the attempt to quantize mathematical objects which are incorrectly given status as physical objects.

I, myself, am looking at an abstract group-theoretic approach. I see the Lie groups as having direct operational meaning as sets of relativity transformations changing the values of observables. Via Noether we then have the observables laid out as generators of the Lie group, and the various manifolds e.g. space-time or Kaluza-Klein space--time--gauge-parameters are simply sub-manifolds of the master relativity group. They are not to be quantized. You rather already have a quantum theory when you select a pseudo-unitary representation, impose the gauge condition a la Gupta-Bleuler by projecting mode vectors onto a positive definite Hilbert space, and thence utilize Born's probability interpretation.

I expect gravitation to emerge naturally once I clearly identify the emergence of a preferred space-time substructure due to a condensation process.

It is quite a departure in formulation from current methods and possibly way too ambitious but I think of it as excising traditional artifacts deriving from pre-relativistic classical mechanics and getting to the heart of the physics. Once you're done with the various adventures into deep mathematics, the quantum physics manifests as the physical interpretation of elements of a Lie algebra as observables and as generators of (passive or active) physical transformations.

One possible tool in my approach is to find a solid meaning for the Hamilton's action principle within my paradigm.

Well I went on a bit about my own ideas, but possibly it will allow you to see why I take the position I do on the parametric nature of space-time coordinates.

Regards,
James Baugh

15. May 27, 2007

### Hans de Vries

One should be able to just use:

$${\bf x}\ =\ \frac{1}{m}\int\ {\bf r} {\bar \psi} \psi$$

As a position operator, or as the center of gravity in case of a multiple particle
representation. You're probably referring to canonical quantization which
doesn't tell us anything about positions at all indeed.

I tend to regard second quantization as an auxiliary tool. Note that you get
quantization for free in the exact non-pertubative QFT solutions in a semi-
classical way.

For instance the exact solution of a Dirac electron in a sinusoidal EM field
is a super position of electrons having absorbed or emitted 0, 1, 2 or more
photons with the frequency of the field. Quantization is guaranteed semi-
classically here.

See for instance the latest paragraph in RQT part 1 of landau and Lifshìtz
around (98.7). An online source which reviews this can be found here:
http://arxiv.org/abs/hep-ph/0408288
(see section 5 and especially equations (53) through (55) )

Regards, Hans

Last edited: May 27, 2007
16. May 27, 2007

### Anonym

Then particle position must be an observable. Study P.A.M. Dirac and E.P.Wigner.

Hi, Hans!

What is your comment on Dirac position operator?

Regards, Dany.

17. May 28, 2007

It is the observer's clock I observe, e.g. by looking at the dials of my watch. However, I cannot do this in darkness; I have to turn on my flashlight. When the photons bounce off my clock, it acquires momentum and undergoes time dilatation, changing the definition of time.

In principle you should see this effect in experiments. Even if you prepare a beam with a perfectly sharp momentum (within the resolution of the experiment), there will still be a broadening of the peaks because the detector recoils and undergoes time dilatation.

There is considable freedom in choosing coordinates; in Minkowski space, they are only defined up to Poincare transformations. However, the distance between two events is something that you can measure with rods and clocks, depending on whether the separation is space- or time-like.

http://www.arxiv.org/abs/hep-th/0701164 and references therein.

The standard canonical formulation involves a foliation of spacetime labelled by a global c-number time. This is the origin of Pauli's theorem and underlies the parametric nature of time in QM. What I am doing is different: introduce canonical commutators for the histories in spacetime, and impose dynamics as a constraint.

I am not doing LQG. My representations are of lowest-energy type.

Alas, you must first construct the right representation. This is nontrivial when you come to groups of diffeomorphisms and gauge transformation in more than 1D. In quantum theory, we want lowest-energy reps, because there some Hamiltonian which is bounded from below. My starting point was the successful construction of lowest-energy reps of the (necessarily anomalous) diffeomorphism algebra in http://www.arxiv.org/abs/math-ph/9810003 .

Note that the standard physicists toolbox is really restricted to algebras living in 1D, i.e. chiral algebras such as the Virasoro and affine Kac-Moody algebras. In order to successfully describe 4D quantum physics, I am convinced that we need their generalizations to 4D, whose representation theory I explained in the paper above.

18. May 29, 2007

### jambaugh

I'm not sure why you say it must follow? Especially from my point. I'm not familiar with any device of physical phenomenon which uniformly translates momentum. Such an act say on a Hydrogen atom would strip the electron from the nucleus and if translating sufficiently would strip the proton from the neutron given their distinct masses.

Now it would follow if you asserted that every canonical transformation as physically actualizable. Or at least that such is the case for the canonical transformations corresponding to the canonical coordinates, and then only if the coordinate is not viewed as a gauge degrees of freedom. I assert that in the canonical formulation the space-time coordinates are specifically gauge degrees of freedom just as is the U(1) phase for gauge EM.

In the general canonical treatment of constrained systems not all functions nor coordinates of the canonical phase-space are observables.
Rather one must apply the gauge constraints to identify a sub-manifold of of states within the whole of the gauge-extended phase space. Through suitable constraints you can make e.g. the initial position and time an observable. However configuration coordinates are no longer necessarily observables.

See e.g: Henneaux and Teitelboim's "Quantization of Gauge Systems".

Recall also that under said constraints the Poisson bracket must be modified, yielding the Dirac bracket. When all is said and done the symplectic phase space is just a particular starting point in defining the dynamic/kinematic Lie algebra (which the Dirac brackets on observables defines) and then the ordinates of said Lie algebra e.g. the generators correspond to the observables and their dual coordinates are parameters.

The Born reciprocity between coordinate is neither necessary nor desired nor always possible in the final analysis. I think it is a hindrance to insist upon it.

WRT "Dirac's Position Operator" you'll have to give me some reference as I'm not familiar with Dirac giving it special attention. But I will say that if you want to treat both space and time coordinates as observables together than that's fine by me. I prefer not to.

But I again say that treating space-time coordinates as observables is neither necessary nor is its absence a problem which only a Bhomian "interpetation" can resolve.

Regards,
Baugh... James Baugh (key Bond theme music!)

19. May 30, 2007

### Anonym

Stone.

Obvious, but they are no longer the particle position and time.

P.A.M.Dirac “The Principles of QM”, fourth edition, Oxford, (1958), paragraph 69. See also E. Schrödinger, Sitzungsber. Preuss. Acad. Wiss., 418, (1930).

It is clear that our attitudes are different and further discussion is pointless (that does not mean that yours is wrong). But let me explain mine.

Here we are at scale of QG. Gravitation has different status in physics. We almost have no experimental data which provide constraints for formulation the theory. And certainly can’t perform the intentional “laboratory” verifications of the theoretical predictions. In hep we have tons.

Consider for example A. Einstein GR. It based solely on one universal experimental result that the inertial mass is identical to the grav. mass. Apparently, A. Einstein was required only to invent the gravitational equivalent of M. Faraday cage (free falling lift). However, he used theoretical constraints. By the theoretical constraints I mean the requirement that your theory will be naturally (axiomatically) consistent and connected with the rest of the theoretical physics.

The same is valid in QT. It seems to me that the attempt to formulate QG without knowledge of consistent relativistic QM is hopeless. This is like to find needle in haystack. We should understand the notion of time and size of time before. To get you hint what I mean: I expect that for example three particle states in QG (if relevant) is described in terms of 512-dim alternative algebra (Kronecker product of three one particle states each spanned with the Cayley algebra).

Regards, Dany.

20. May 31, 2007

### jambaugh

Thanks for the ref. I have it in hand. Again I don't see where he has given it any special emphasis. He rather uses the observation of position in a classical sense to determine an average velocity, and to argue informally via the uncertainty principle that the eigen-values of +or-c for instantaneous velocity is meaningful.
Sorry don't have his book.
Yes, we'll not convince each other but I don't think further discussion is fully pointless. As you say below. This difference of opinion may have an effect of approaches to a QG theory, something it seems we both seek.
We do have quite a bit of empirical data confirming aspects of the GR and so matching the classical theory up to the order of said astronomical phenomena would be something. But you are quite right, unless a new theory could suggest some non-classical phenomenon within the realm of the laboratory experment. Maybe a GRASER?
Quite right. I couldn't agree more. (Though we may disagree w.r.t. of what that knowledge consists.) It has been much on my mind of late. You see my research suggests that a correct consistent theory must allow a (very large but) finite dimensional representation space (pseudo-Hilbert or pHilbert space). This necessarily precludes a unitary representation but not a pseudo-unitary one (as e.g. spinor reps). I have reluctantly accepted the Gupta-Bleuler prescription of projecting mode ("state") vectors onto a positive definite Hilbert subspace to allow consistent probability interpretation, said projecting being a frame dependent (gauge) condition. It of course requires a bit of careful though about interpretation but it is --so far as I've been able to evaluate-- fully consistent, matches the non-relativistic limit and encompasses all the physically actualizable experimental outcomes in its application to standard problems. I've tried a few alternatives e.g. allowing the square amplitude to represent ratios of frequencies and thus not requiring normalizable bounded values. But alas had no luck.

Here I don't agree. As I say I think we have a perfectly good relativistic QM with full operational interpretation. You may disagree with it but it is there. With regard to understanding the "notion and size of time" I agree with the notion but don't think "size" has meaning as again my position is that time is a parametric variable external to the system. You can see without agreeing that such a position "solves" the issue.

Where I think we differ is in the interpretation of Einstein's equivalence principle in so far as it will be applied in a quantum setting. I see the equivalence as saying "geometry is really just an observer-defined component of dynamics" i.e. gauge whereas I'm guessing you take the standard wording that "gravitational dynamics is just geometry".

Where I see this disagreement as making a difference is in attempts to "quantize the space-time manifold". I see this as quantizing what isn't actually real in the sense that an electron is. If I'm wrong then the "Brane-iacs" may yet find that ultimate theory. Only time will tell. But based on the position I've posted here I think their approach and others which attempts to "quantize space-time" are futile.

Have you checked out Tony Smith's ponderings?
http://www.valdostamuseum.org/hamsmith/TShome.html" [Broken]
Brilliant guy, I know him, but I think he should separate his theology from his theory, something he stubbornly refuses to do and so has a hard time being taken seriously. He however makes a very good case for his prediction of top quark mass being more consistent with the experimental data then the accepted value.

Well, as you say we're probably not going to make any headway discussing this further here.

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