Those who use relativistic mass and why

[Garth]

The Klein-Gordon equation that was referred to before is the relativistic version of QM for spinless particles, and it sits with SR just fine.

Apart from the problem of defining the mass by the Klein-Gordon equation of spin 1/2 particles such as an electron, the question is whether QM sits with GR just fine. I think you will find that the problem in developing a quantum-gravitational theory is that QM requires the preferred foliation of space-time referred to above. There may be confusion here; all along I have not been questioning SR but GR and its problems with defining time and mass. If I may repeat the first of my "questions" see "Questions of the equivalence principle" https://www.physicsforums.com/showthread.php?t=32285.
"1. In the presence of gravitational fields the Einstein
Equivalence Principle (EEP) is a necessary and sufficient condition
for the Principle of Relativity, (PR). Here I summarise PR as the
doctrine of no preferred frames of reference. In the absence of such
fields the EEP becomes meaningless, although then the PR does come
into its own and is appropriate in Special Relativity (SR), which was
formulated for such an idealised case. However, if we now re-
introduce gravitational fields, i.e. gravitating masses, do we not
then find that the PR collapses? For in that case is it not possible
to identify preferred frames of reference? Such frames being those of
the Centre of Mass (CoM) of the system in question and the universe
as a whole, (that in which the Cosmic Microwave Background is
globally isotropic.) The CoM is preferred in the sense that only in
that frame of reference, that is the centroid measured in the frame
co-moving with the massive system, is energy conserved as well as
energy-momentum. But if the PR is not valid in the presence of
gravitational masses then surely the EEP cannot be either? "

 

[quantumdude]

Apart from the problem of defining the mass by the Klein-Gordon equation of spin 1/2 particles such as an electron, the question is whether QM sits with GR just fine.

First, there is no conflict between the definition of mass as an invariant and the KG equation. The KG equation is just the quantized version of the relation E²-p²=m².

Second, the KG equation does apply to spin-1/2 particles. It just applies to components of the Dirac wavefunction.

Third, the Dirac equation, which is the equation that describes the quantum mechanics of spin-1/2 particles, is also Lorentz covariant, and it also sits just fine with SR.

And finally, there is no conflict between QM and GR. There is nothing at all stopping one from doing QM in curved spacetime. The problem comes into play when one tries to come up with a quantum theory of gravity, but that's not QM.

I think you will find that the problem in developing a quantum-gravitational theory is that QM requires the preferred foliation of space-time referred to above.

But QM doesn't require a preferred frame of reference. Relativistic QM is just that: QM that conforms to SR.

 

[DW]

Thank you for that; in which case not only is the wavelength statistical in nature (as with the rest of q-m) but also the argument is circular;

No it isn't.

the wavelength is defined in terms of the particle's mass

No it isn't.

Do not quantum mechanical definitions require a preferred foliation of space-time, a preferred frame of reference - normally that of the observer?

No.

 

[pervect]

ISuch frames being those of
the Centre of Mass (CoM) of the system in question and the universe
as a whole, (that in which the Cosmic Microwave Background is
globally isotropic.) The CoM is preferred in the sense that only in
that frame of reference, that is the centroid measured in the frame
co-moving with the massive system, is energy conserved as well as
energy-momentum. But if the PR is not valid in the presence of
gravitational masses then surely the EEP cannot be either? "

There isn't any frame of reference in which the CMB is globally isotropic. If you pick a point, there's a local frame in which the CMB is isotropic. But it's a different frame at each point, i.e. the frame in which the CMB is isotropic at point A is moving with respect to the frame in which the CMB is isotropic at point B.

I don't understand why you think the CoM frame is special for the conservation of energy-momentum. AFAIK, to define a conserved energy, you need either a local timelike symmetry (a timelike Killing vector), or an asymptotically flat space-time. I've never read about any requirement to be in the center of mass "frame", though it's certainly convenient to calculate in.

I don't follow your argument about the EEP either, but that's probably my own failing. Though I do note that you seem to reject the EEP for the same reasons you require it, which makes me doubt your argument.

 

[Garth]

Tom - Thank you for your comments, I am approaching the QM/GR interface from the GR side and I value your constructive criticisms in order to deepen my understanding.

I am not suggesting there is a conflict between QM and SR, indeed QM might be seen to be derived from SR through the resolution of the "de' Broglie paradox".

I was interested in your remark that the KG equation applied to spin-1/2 particles; my understanding was that as it involves the second time derivative of the Psi state vector the probability density associated with its solutions is not positive definite and therefore it could not represent such particles.

Also your affirmation that “there is no conflict between QM and GR” I find debateable. For example, the statement "No prediction of spacetime, therefore no meaning for spacetime is the verdict of the Quantum Principle. That object which is central to all of Classical General Relativity, the four dimensional spacetime geometry, simply does not exist, except in a classical approximation." (Misner, Thorne and Wheeler, Gravitation, p. 1183) would suggest otherwise.

I appreciate it depends on whether you approach the subject from the QM or the GR side but any problems of either approach should be mirrored in the other I would have thought. I would suggest another such problem to be the energy density and hence curvature associated with the quantum vacuum.

Pervect – Thank you too for your comments.

I appreciate the CMB is globally isotropic in a different frame at each point. I was taking that as understood.

I don’t think the CoM frame is special for the conservation of energy-momentum, which is conserved in all inertial frames as a consequence of the EEP, it is the conservation of energy that is the crucial point.

Finally it is GR that requires the EEP.

 

[quantumdude]

I am not suggesting there is a conflict between QM and SR, indeed QM might be seen to be derived from SR through the resolution of the "de' Broglie paradox".

I don't know what the deBroglie paradox is, but I do know that you cannot derive QM from SR. Schrodinger's QM contradicts SR, inasmuch as the expectation values of observables do not satisfy the classical relativistic Hamiltonian.

I was interested in your remark that the KG equation applied to spin-1/2 particles; my understanding was that as it involves the second time derivative of the Psi state vector the probability density associated with its solutions is not positive definite and therefore it could not represent such particles.

Not at all. While it's true that the problem of non-positive definiteness of the KG probability density r originally was thought to be fatal to the theory, a reinterpretation (by Pauli? can't remember whom) of r as a charge density qr solved the problem. Charge densities are not required to be positive definite.

Also your affirmation that “there is no conflict between QM and GR” I find debateable.

It's not debatable, general relativistic quantum mechanics exists. It's just not found in textbooks in a standard graduate curriculum because it's so specialized. All you have to do is replace the SR metric with the GR metric of your choice, and start calculating. The theory is just as well-defined as KG or Dirac.

 

[quantumdude]

I neglected to comment on the following important point.

That object which is central to all of Classical General Relativity, the four dimensional spacetime geometry, simply does not exist, except in a classical approximation." (Misner, Thorne and Wheeler, Gravitation, p. 1183) would suggest otherwise.

You seem to be consistently mixing up the ideas of QM, QFT, and Quantum Gravity. When people say that "GR and quantum theory are not compatible", they mean that a theory of quantum gravity doesn't exist. They do not mean that GR is inconsistent with QM or QFT. Indeed, QM and QFT can be done in a curved spacetime.

When people talk about "quantization", they refer to the quantization of 3 things: dynamical variables, fields, and spacetime (the metric). The quote from MTW is referring to the latter type. But the metric in QM and QFT is perfectly classical. It's the metric in Quantum Gravity that is not.

Here's a summary:

Quantum Mechanics
Dynamical variables: quantized
Fields: classical
Spacetime: classical

Quantum Field Theory
Dynamical variables: quantized
Fields: quantized
Spacetime: classical

Quantum Gravity
Dynamical variables: quantized
Fields: quantized
Spacetime: quantized

 

[Garth]

Tom, thank you again.

I don't know what the deBroglie paradox is, but I do know that you cannot derive QM from SR. Schrodinger's QM contradicts SR, inasmuch as the expectation values of observables do not satisfy the classical relativistic Hamiltonian.

I understand that in 1923 de Broglie's paradox was one of Schrodinger's starting points in his formulation of QM. The paradox is that between on the one hand the de Broglie frequency, which is derived from the de Broglie wavelength and which is proportional to total energy and therefore increases with relative velocity and on the other hand the observed frequency of a moving clock which decreases with relative velocity. I believe it was the recognition that these were two totally different frequencies that led Schrodinger to develop the idea of the de Broglie's 'wave' into a wave function. However both the de Broglie's 'wave' and the wave function are not observables themselves but may be used to predict such.

Not at all. While it's true that the problem of non-positive definiteness of the KG probability density r originally was thought to be fatal to the theory, a reinterpretation (by Pauli? can't remember whom) of r as a charge density qr solved the problem. Charge densities are not required to be positive definite.

Thank you I live and learn!

It's not debatable, general relativistic quantum mechanics exists. It's just not found in textbooks in a standard graduate curriculum because it's so specialized. All you have to do is replace the SR metric with the GR metric of your choice, and start calculating. The theory is just as well-defined as KG or Dirac.

"the metric of your choice" in which frame may I ask, a preferred one perhaps? Or does general relativistic quantum mechanics work for any inertial frame in general?

 

[pervect]

I don’t think the CoM frame is special for the conservation of energy-momentum, which is conserved in all inertial frames as a consequence of the EEP, it is the conservation of energy that is the crucial point.

But the conservation of energy-momentum implies the conservation of energy. For a closed system, the conservation of energy-momentum means that one has four quantites which are conserved by the system as it evolves in time - these quantities are E, a scalar, and the three components of P.

You are apparently using some different defintion. You seem to be demanding that energy by conserved over some sort of transformation of the system (a Lorentz Boost?). This isn't the right definition.

I think this has been pointed out before, even in classical mechanics, or special relativity, there is no guarnatee that different observers will compute the same value of energy for a closed system, there is only the guarantee that each observer will find that the energy of the closed system is constant as (their) time evolves.

Anyway, to be able to define the conserved energy of a closed system in GR, one needs either a local timelike symmetry, in which case the conservation of energy is obvious by Noether's theorem, or one needs an asymptotically flat space time.

 

[Garth]

You seem to be consistently mixing up the ideas of QM, QFT, and Quantum Gravity.

Tom - Indeed I have been talking about Quantum Gravity, sorry about the confusion and thank you for the clarification.


Pervect - It was Noether's second theorem that demonstrated that GR was a type of "improper energy theorem" and that in general it did not, and would not be expected to, conserve energy.
I am thinking about a single observer in an inertial frame freely falling towards a central gravitational mass, the Earth for example. That observer would conclude that the Earth was accelerating towards him even though both he and the Earth were in free fall with no (first order) gravitational forces acting, just the convergence of their geodesics over curved spacetime. In the observer's frame of reference the components of the metric would be time dependent and hence the time component of the Earth's energy momentum vector, its total energy, would not be conserved.

 

[quantumdude]

I understand that in 1923 de Broglie's paradox was one of Schrodinger's starting points in his formulation of QM. The paradox is that between on the one hand the de Broglie frequency, which is derived from the de Broglie wavelength and which is proportional to total energy and therefore increases with relative velocity and on the other hand the observed frequency of a moving clock which decreases with relative velocity. I believe it was the recognition that these were two totally different frequencies that led Schrodinger to develop the idea of the de Broglie's 'wave' into a wave function. However both the de Broglie's 'wave' and the wave function are not observables themselves but may be used to predict such.

I still don't see the paradox. The frequency of timekeeping the moving clock is not in any way related to the deBroglie frequency. But the deBroglie frequency of the moving clock (the frequency of the matter wave associated with the moving clock) varies with momentum just as QM says it should.

"the metric of your choice" in which frame may I ask, a preferred one perhaps? Or does general relativistic quantum mechanics work for any inertial frame in general?

Metrics don't vary from frame to frame. Metrics are what tell you how to transform from frame to frame. When I say "metric of your choice", I mean equivalently "energy momentum tensor of your choice". That is, near a stationary black hole, you will have one metric. Near a spinning black hole, you'll have another. Near a charged spinning black hole, you'll have still another.

But for any given metric, it is possible to formulate a generally covariant quantum mechanics in that spacetime.

 

[Haelfix]

Just to specify, the KG equation works for spin 1/2 particles, but doesn't encapsulate the full theory. The Dirac equation is more general in that context as it contains the equations of motion of antiparticles in addition.

And then qft generalizes that one step further with full Grassmann mechanics for fermions and the polarization of the vacuum.

The problem with Quantum field theory in curved spacetime is fivefold IMO.

1) Its a nonrenormalizable theory (which is fine in the modern context, it just means that its not the full story)
2) Its hard to find meaningfull local observables in the theory (which is troubling), indeed perhaps only the S Matrix makes sense.
3) Its conceptually hard to take into account back reactions from the geometry, particularly so in very strong curvature regimes.
4) Half of QFT becomes intrinsically problematic, unless you believe in the Euclidean path integral being applicable. You really need to work in the Hamiltonian context, but then you have to figure out clever ways to foliate spacetime so your configuration space is meaningfull. Its best to work in the algebraic framework in this context.
5) Weird things show up, like conformal anomalies and the measurement problem is further complicated. There are solutions to this however, but the technical details are still debated about in the specialist circles.

But the theory seems to work fairly well albeit being a difficult subject with lots of unpleasant results and some unbeautiful brute force methodology. Its just been a little dated since String theory and the other quantum gravity theories have surfaced, which look at the problem more from the bottom-up, and seem to generalize things several steps further (and presumably remove all ambiguities in that limit).

 

[Garth]

I still don't see the paradox.

That's the point, Schrodinger resolved it by formulating his representation of QM. However prior to QM it was a paradox, the study of which inspired/informed Schrodinger to make his conceptual leap.

Metrics don't vary from frame to frame.

Again my apologies I sometimes make mistakes reading from the screen and I read "frame" when you had written "metric". I am interested in the problem of time in canonical quantum gravity.
Haelfix - Thank you, being a relativist I am a stranger straying into another discipline in QFT, seeking understanding. I was aware there were problems on the interface between the two disciplines but were unsure of what they were exactly.

"Why keep your mouth closed so as not to appear a fool when you can open it and prove you are!"

 

[pmb_phy]

Isn't this getting a bit off topic? Perhaps you can start a new thread in the appropriate forum so that those interested in the current suibject (i.e. QM) can participate or follow along. If you keep it here then there is little reason to assume that QM people would read this thread.

Just a suggestion mind you.

Pete

 

[Garth]

Opening my mouth again (!), the present discussion may be on topic if it is (as it started out) a consideration of how mass may be defined in the various theories and conventions. At a fundamental level, if mass is the energy of string vibrations, de Broglie waves, or whatever, then the distinction between mass and energy maintained by the "mass is invariant" convention breaks down; it is all a sea of energies and virtual particles transmitting forces. If so at the most fundamental level, then perhaps for consistency sake, might it be thought of as such at higher levels?

 

[pmb_phy]

Opening my mouth again (!), the present discussion may be on topic if it is (as it started out) a consideration of how mass may be defined in the various theories and conventions.

That is not the topic of this thread. The topic is Those who use relativistic mass and why

Pete

 

[pmb_phy]

I never realized it before today but Wald does use the term "mass" in at least one place to refer to "relativistic mass" in his text General Relativity.

If you have his text see page 62 right below Eq. (4.2.15). He speaks of conservation of mass. What he's referring to is the conservation of inertial energy which is proportional to (relativistic aka inertial) mass.

Pete

 

[DW]

I never realized it before today but Wald does use the term "mass" in at least one place to refer to "relativistic mass" in his text General Relativity.

If you have his text see page 62 right below Eq. (4.2.15). He speaks of conservation of mass. What he's referring to is the conservation of inertial energy (i.e. T⁰⁰) which is proportional to relativistic mass.

Pete

Not only is T⁰⁰ NOT mass, but it is NOT energy either. It is an energy density.

 

[pervect]

I never realized it before today but Wald does use the term "mass" in at least one place to refer to "relativistic mass" in his text General Relativity.

If you have his text see page 62 right below Eq. (4.2.15). He speaks of conservation of mass. What he's referring to is the conservation of inertial energy which is proportional to (relativistic aka inertial) mass.

Pete

I read this differently. I think that Wald was saying that if we consider a perfect fluid, and take the equations of motion $$\partial^{a}T_{ab}=0$$, one of the unsurprising results in the non-relativsitic limit is that the mass of the perfect fluid is conserved.

 

[pmb_phy]

I read this differently. I think that Wald was saying that if we consider a perfect fluid, and take the equations of motion $$\partial^{a}T_{ab}=0$$, one of the unsurprising results in the non-relativsitic limit is that the mass of the perfect fluid is conserved.

Thanks. That is what Wald says. My mistake. Thanks for pointing that out. I was flipping through Wald and saw that and it seemed to agree with what I posted here in Eq. (1) - http://www.geocities.com/physics_world/sr/mass_tensor.htm

Note: There are other authors who do use the term "mass" to mean "relativistic mass". See example listed above, i.e. Cosmological Physics, John A. Peacock, page 18

The only ingredient now missing from a classical theory of relativistic gravitation is a field equation: the presence of mass must determine the gravitational field. [...] Now, if this equation is to be covariant, T^uv must be a tensor and is known as the energy-momentum tensor (or sometimes as the stress-energy tensor). The meanings of its components in words are T^00 = c^2x(mass density) = energy density, T^12 = x-component of current of y-momentum etc. From these definitionsl the tensor is readily seen to be symmetric. Both momentum density and energy flux density are the product of a mass density and a net velocity, so T^0m = T^m0.

By "mass density" he means what you'd call "relativistic mass density" and what Wald(page 60)/MTW call "mass-energy density."

Pete