Conserved quantities under the Lorentz boost

[ncarron]

In physics, a symmetry of the physical system is always associated with some conserved quantity.
That physical laws are invariant under the observer’s displacement in position leads to conservation of momentum.
Invariance under rotation leads to conservation of angular momentum, and under displacement in time leads to conservation of energy.
However, there is one more symmetry: That physical laws are invariant when the entire system is moving at a constant velocity relative to a first system (Galilean or Lorentz invariance; Lorentz "boost"). What is the associated conserved quantity under that symmetry?

 

[meopemuk]

The following combination of observables is conserved: $HR - Pc^2t$ where $H$ is the energy, $R$ is the center-of-mass position, and $P$ is the momentum. In other words, this means that the center of mass of any system moves with constant velocity along a straight line

$R(t) = R(0) + (Pc^2/H)t$

where $Pc^2/H$ is the relativistic definition of velocity. This is an independent conservation law.

Eugene.

 

[ncarron]

Eugene, thanks for the reply.

The momentum and energy conservation laws mean the total momentum $P$ and the total energy $H$ are constant in time. Therefore $V==Pc^2/H$ is a constant. So it seems the first two conservation laws already imply that the center of mass moves at a constant velocity. And yet you state "This is an independent conservation law". Can you explain?

My original question seems like one that should be discussed in physics textbooks. But I have not found one. Do you know of any text(s) in which it is discussed?

- Neal

 

[meopemuk]

Conservation of momentum means $P=const$.
Conservation of energy means $H=const$.
But these two conservation laws can't tell us what's the time dependence of the center of mass. So, we need a separate conservation law $HR-Pc^2t=const$.

Usually, the connections between symmetries and conservation laws are discussed within the context of the Noether theorem. See the Wikipedia article "Noether's theorem", "Example 2. Conservation of center of momentum."

However, I prefer a more fundamental approach based on unitary representations of the Poincar'e group in quantum mechanics. This approach was initiated by two seminal articles

E. P. Wigner, "On unitary representations of the inhomogeneous Lorentz group", Ann. Math. 40 (1939), 149.

P. A. M. Dirac, "Forms of relativistic dynamics", Rev. Mod. Phys. 21 (1949) 392.

Unfortunately, these ideas and their relations to conservation laws do not have a proper coverage in textbooks. You can try a couple of places:

A. O. Barut, R. Raczka, "Theory of group representations and applications" (Polish Scientific Publishers, Warszawa, 1980), Chapter 13.

E. Stefanovich, "Elementary particle theory. vol. 1. Quantum mechanics" (de Gruyter, Berlin, 2018), Chapter 4.

Eugene.

 

[ncarron]

I assume $P = \gamma M V$ , and $H = \gamma M c^2$, where $M$ is the total rest-mass of the entire system, $u$ is the given velocity of relative motion of the reference frames, and $\gamma = 1/\sqrt{1-(u/c)^2}$. The "rest-mass of the entire system" is the mass-energy of the entire system in a frame in which the CoM is at rest.

If the constant $P c^2 / H = V$ is not the velocity of the CoM, what is it the velocity of?

If a separate, independent conservation law is needed for the velocity of the CoM (and conservation of $P$ and of $H$ alone do not guarantee the CoM moves at a constant velocity), it seems to imply that even if $P$ and $H$ are constant, the velocity of CoM may not be. Can you provide a physical example in which $P$ and $H$ are constant but the velocity of CoM varies in time?

 

[meopemuk]

Let me try from a slightly different angle.

We have a 10-parameter Poincar'e group of inertial transformations for which the relativity principle can be applied. This group should be represented by unitary operators in the Hilbert space of any isolated quantum system. The 10 independent generators of this representation are Hermitian operators, which we identify with total observables of the system. They are the Hamiltonian $H$, the total momentum vector $\boldsymbol{P}$, the total angular momentum vector $\boldsymbol{J}$, and the boost operator $\boldsymbol{K}$.

From commutation relations of the Poincar'e Lie algebra, we can find out how the above total observables transform with respect to time translations. The first three operators commute with the Hamiltonian $[H,H]=[\boldsymbol{P}, H] = [\boldsymbol{J}, H]=0$, so we get the 7 conservation laws immediately. We can also define an operator

$\boldsymbol{V} \equiv \boldsymbol{P}c^2/H$, (1)

call it "velocity," and verify that it is conserved. But at this point there is nothing relating this operator to the observable of position. Some extra logical steps have to be taken to establish this relationship.

Now we turn our attention to the boost operator, which does not commute with the Hamiltonian: $[\boldsymbol{K}, H] = -i \hbar \boldsymbol{P}$. This means that, strictly speaking, there is no conservation law associated with boosts. However, $\boldsymbol{K}$ depends on time linearly

$\boldsymbol{K}(t) = \boldsymbol{K}(0) + \boldsymbol{P}t$, (2)

so we can say that there is an independent law stating that "the center-of-mass $\boldsymbol{K}$ is moving uniformly."

Now, it is not common to use the boost observable $\boldsymbol{K}$ in physics. Traditionally, we use the better known "center-of-mass position", which is related to the boost observable by formula $\boldsymbol{R} \equiv - \boldsymbol{K}c^2/H$. (This is the famous Newton-Wigner formula, where I omitted spin contributions for simplicity and pretended that $\boldsymbol{K}$ and $H$ commute.) Only now we can use eqs. (1) and (2) and confidently say that the center-of-mass position of any physical system moves uniformly and that the rate of this motion is given by the (conserved) velocity operator introduced earlier

$\boldsymbol{R}(t) = \boldsymbol{R}(0) + \boldsymbol{V}t$

I hope this answers your original question about the conservation law associated with boosts.

Eugene.

 

[haushofer]

Likewise, in classical mechanics the conserved charge under Galilei boosts is, in my experience, often not well-covered. Or maybe I missed it.

 

[meopemuk]

IMHO, the best discussion of relativistic (both Galilei and Poincar'e) symmetries in classical mechanics can be found in

E. C. G. Sudarshan, N. Mukunda, "Classical dynamics: A modern perspective", (John Wiley & Sons, New York, 1974). Chapter 20.

Eugene.

 

[ncarron]

Eugene,

My natural bent is rather more physical than mathematical. Hence the tone of my previous question(s).

Your last reply in terms of the Poincare group does answer the question, but is more formal than I am familiar with. I will have to study both the Lorentz and Poincare groups.

Thank you for your thorough replies.

- Neal

 

[meopemuk]

Hi Neal,
I strongly recommend you to look into the Poincare group and its unitary representations. This is a piece of math, which is densely packed with physical applications. Once you learn this tool, you'll know everything about relativistic quantum mechanics (and a big portion of QFT):

1. Definitions of physical observables,
2. Their commutation relations,
3. Conservation laws,
4. Transformation properties,
5. Classification of particles by their mass and spin,
6. Structure of Hilbert spaces: one-particle, multiparticle, Fock space.
7. Definitions of relativistic interactions (Dirac's forms of dynamics).

I would dare to say that the entire field of theoretical particle physics is nothing but an application of the theory of unitary representations of the Poincare group.

Eugene.

 

[ncarron]

Eugene,
That sounds like an excellent suggestion. I have downloaded Wigner's and Dirac's papers, but I imagine the Lorentz and Poincare groups are discussed in many review articles and textbooks as well.
- Neal

 

[meopemuk]

Yes, there are lot of articles, but, unfortunately, no single comprehensive review that would cover all aspects. The best you can do is to use Google Scholar to see who cited the above papers by Wigner and Dirac.

One important reference is
S. Weinberg, "The Quantum Theory of Fields, Vol. 1", (University Press, Cambridge, 1995).

He covers Wigner's classification of particles by mass and spin and then makes a smooth transition to quantum fields. Priceless!

Eugene.