Matt Strassler popularizes the short preprint by Andrew Cohen and Sheldon Glashow who argued that superluminal neutrinos would emit a Čerenkov-like Bremsstrahlung radiation composed of charged lepton pairs which would dramatically influence their behavior.
The first commenter under Matt's article is... well, his name is banned on this blog so don't try to write any of these two obscene words in the comments. He boldly claims that the "second most likely" explanation of the OPERA experiment – after the first most likely explanation that there is some error and the actual speed is not higher than the speed of light – is some kind of "deformation of special relativity without a privileged reference frame".
What we're really talking about is the κ-Poincaré algebra; using my international keyboard, the Greek letter "kappa" is written as "accuted zero" and you may write hundreds of other characters, too.
The κ-Poincaré algebra is a mathematically interesting deformation of the Poincaré algebra that was written down by Shahn Majid and Henri Ruegg in 1994. So let me emphasize at the very beginning that the loop-quantum-gravity-like crackpots have made zero contribution to this mathematical physics and even though they adopted the κ-Poincaré algebra as a talking point to justify some of the loop-quantum-gravity delusions, there doesn't exist a single argument that would rationally connect the κ-Poincaré algebra with loop quantum gravity (or any similar nonsensical approach to quantum gravity). In particular, it's obvious that the loop-quantum-gravity-like violations of the Lorentz symmetry do pick a privileged frame and destroy relativity in the most brutal, generic way.
Just to be sure, the Poincaré algebra is the Lie algebra producing infinitesimal transformations of the Poincaré group. The Poincaré group is the combination of the Lorentz transformations underlying the special theory of relativity, including the rotations, and the translations of the Minkowski spacetime that is generated by the momenta.
The letter κ means that it is a quantum group, or a quantum deformation of the original Lie algebra. Such quantum deformations of Lie groups are mathematically classified as "Hopf algebras". The letter κ itself is a mathematical variable. If you choose κ=0, you obtain the original Poincaré algebra. From this viewpoint, the terminology κ-Poincaré is analogous to q-deformed algebras; the letter q is a mathematical variable, too.
In advance, I would like to say that I consider quantum groups to be a very natural and arguably important part of the mathematical physics cannon (of quantum field theories) in at most 1+1 spacetime dimensions. However, they become very unnatural in more than 1+1 and especially more than 2+1 dimensions because one expects some kind of locality in 3+1 dimensions and higher.
As we will see, the notion of locality is severely crippled by the quantum deformation. In 1+1 dimensions, e.g. on the world sheet of strings, locality doesn't have far-reaching consequences because in 1 spatial dimension (in the Lineland), there's not enough "room" for things to be "truly independent" of each other (particles are equivalent to semi-infinite kinks, and so on): so quantum groups may be very important in 1+1 dimensions. But the higher spacetime dimension you consider, the more natural it is to talk about locality, the more important role for the laws of physics is played by locality (or approximate locality), and the more unnatural theories that conflict with locality become.
Wikipedia which renames the letter κ as λ (lambda, accuted left square bracket), gives us the following commutation relations for the κ-Poincaré algebra (let us start with those that are not affected by the κ parameter):
\[ [P_\mu, P_\nu] = 0. \] That was easy, the momenta (and translations) commute with each other. Also,
\[ [R_j , P_0] = 0. \] That was also easy: the rotations commute with the energy.
\[ [R_j , P_k] = i \varepsilon_{jkl} P_l \] The momenta transform as vectors under rotations
\[ [R_j , N_k] = i \varepsilon_{jkl} N_l \] and so do the boost generators \(N_k\). Moreover, the commutator of two rotation generators or two boost generators is a rotation generator, just like in the ordinary Poincaré algebra:
\[ [R_j , R_k] = i \varepsilon_{jkl} R_l, \quad [N_j,N_k] = -i \varepsilon_{jkl} R_l \] Also, the boosts transform the energy into a spatial component of the momentum, just like you expect:
\[ [N_j , P_0] = i P_j \] But this is where your old knowledge ends. The remaining commutator does depend on κ:
\[ [N_j , P_k] = i \delta_{jk} \left( \frac{1 - e^{- 2 \lambda P_0}}{2 \lambda} + \frac{ \lambda }{2} |\vec{P}|^2 \right) - i \lambda P_j P_k \] The momenta are transformed under boosts in a nonlinear way we will discuss. Note that in the \(\lambda\to 0\) limit, you get the usual \(i\delta_{jk}P_0\).
The nonlinear function of the momentum generators on the right hand side looks arbitrary but it is not really arbitrary: the Hopf algebras have kind of constrained rules and this structure is forced upon you. Alternatively, you obtain very analogous nonlinear commutation relations when you study various important 2-dimensional solvable field theories that have "quantum groups" as their symmetries.
Note that if the energy and momenta \(P_0,P_i\) are conserved in one frame, they will be conserved in a frame boosted by the \(N_i\) generators as well. But the energy and momenta in the new frame will be a nonlinear function of the original ones. As I tried to explain more than five years ago, in Locality and additivity of energy, such a nonlinear formula for the total energy and the total momentum spoils locality.
Imagine that you have two subsystems in different parts of the Milky Way. Unless it is possible to send instantaneous signals, the two subsystems should evolve independently of one another. In the quantum setup, this means that the state describing both objects must approximately be equal to the tensor product of states describing each of them,
\[ |\psi\rangle = |\psi_1\rangle \otimes |\psi_2\rangle \] while the Hamiltonian acting on the overall state \(|\psi\rangle\) must be simply a sum
\[ H = H_1 + H_2 \] of the Hamiltonians acting on the subsystems. This is necessary for the independence of the systems (the tensor factorization of the overall state) to be preserved under time evolution. So the energy should be additive: each of the two systems should simply add the right term. For similar reasons, the total momentum must be the sum of the momenta of the distant pieces as well.
But if this additivity for the energy and momentum holds in one frame, it will inevitably be violated in another frame that you obtain by boosts generated by \(N_i\). Indeed, under these κ-boosts, the energy and momentum transform to
\[ (E,p_i)\to {\rm nonlinear\,\,gibberish} (E,p_i) \] which is too bad because nonlinear gibberish functions of the total \(E\) and the total \(p_i\) are obviously not sums of the pieces coming from the two places of the Milky Way.
There exist all kinds of mixings which prove that even if the locality of the laws of physics were valid in one reference frame, it won't hold in any other κ-deformed reference frame. The evolution of the object on our side of the Milky Way will strongly depend on all kinds of products of momenta of our object and the object on the other side of our Galaxy. That's too bad. One may show that different observers (who are boosted relatively to each other) will also inevitably disagree on whether or not an object is caught inside a box, and so on.
To summarize, decent laws of physics in more than 1+1 dimensions prohibit some brutal forms of nonlocal communication and respect some laws of locality, at least approximate ones. That's why theories with quantum-deformed symmetries are almost certainly physically uninteresting for our understanding of dynamics in spacetimes or world volumes above 1+1 dimensions.
And that's the memo.
The first commenter under Matt's article is... well, his name is banned on this blog so don't try to write any of these two obscene words in the comments. He boldly claims that the "second most likely" explanation of the OPERA experiment – after the first most likely explanation that there is some error and the actual speed is not higher than the speed of light – is some kind of "deformation of special relativity without a privileged reference frame".
What we're really talking about is the κ-Poincaré algebra; using my international keyboard, the Greek letter "kappa" is written as "accuted zero" and you may write hundreds of other characters, too.
The κ-Poincaré algebra is a mathematically interesting deformation of the Poincaré algebra that was written down by Shahn Majid and Henri Ruegg in 1994. So let me emphasize at the very beginning that the loop-quantum-gravity-like crackpots have made zero contribution to this mathematical physics and even though they adopted the κ-Poincaré algebra as a talking point to justify some of the loop-quantum-gravity delusions, there doesn't exist a single argument that would rationally connect the κ-Poincaré algebra with loop quantum gravity (or any similar nonsensical approach to quantum gravity). In particular, it's obvious that the loop-quantum-gravity-like violations of the Lorentz symmetry do pick a privileged frame and destroy relativity in the most brutal, generic way.
Just to be sure, the Poincaré algebra is the Lie algebra producing infinitesimal transformations of the Poincaré group. The Poincaré group is the combination of the Lorentz transformations underlying the special theory of relativity, including the rotations, and the translations of the Minkowski spacetime that is generated by the momenta.
The letter κ means that it is a quantum group, or a quantum deformation of the original Lie algebra. Such quantum deformations of Lie groups are mathematically classified as "Hopf algebras". The letter κ itself is a mathematical variable. If you choose κ=0, you obtain the original Poincaré algebra. From this viewpoint, the terminology κ-Poincaré is analogous to q-deformed algebras; the letter q is a mathematical variable, too.
In advance, I would like to say that I consider quantum groups to be a very natural and arguably important part of the mathematical physics cannon (of quantum field theories) in at most 1+1 spacetime dimensions. However, they become very unnatural in more than 1+1 and especially more than 2+1 dimensions because one expects some kind of locality in 3+1 dimensions and higher.
As we will see, the notion of locality is severely crippled by the quantum deformation. In 1+1 dimensions, e.g. on the world sheet of strings, locality doesn't have far-reaching consequences because in 1 spatial dimension (in the Lineland), there's not enough "room" for things to be "truly independent" of each other (particles are equivalent to semi-infinite kinks, and so on): so quantum groups may be very important in 1+1 dimensions. But the higher spacetime dimension you consider, the more natural it is to talk about locality, the more important role for the laws of physics is played by locality (or approximate locality), and the more unnatural theories that conflict with locality become.
Wikipedia which renames the letter κ as λ (lambda, accuted left square bracket), gives us the following commutation relations for the κ-Poincaré algebra (let us start with those that are not affected by the κ parameter):
\[ [P_\mu, P_\nu] = 0. \] That was easy, the momenta (and translations) commute with each other. Also,
\[ [R_j , P_0] = 0. \] That was also easy: the rotations commute with the energy.
\[ [R_j , P_k] = i \varepsilon_{jkl} P_l \] The momenta transform as vectors under rotations
\[ [R_j , N_k] = i \varepsilon_{jkl} N_l \] and so do the boost generators \(N_k\). Moreover, the commutator of two rotation generators or two boost generators is a rotation generator, just like in the ordinary Poincaré algebra:
\[ [R_j , R_k] = i \varepsilon_{jkl} R_l, \quad [N_j,N_k] = -i \varepsilon_{jkl} R_l \] Also, the boosts transform the energy into a spatial component of the momentum, just like you expect:
\[ [N_j , P_0] = i P_j \] But this is where your old knowledge ends. The remaining commutator does depend on κ:
\[ [N_j , P_k] = i \delta_{jk} \left( \frac{1 - e^{- 2 \lambda P_0}}{2 \lambda} + \frac{ \lambda }{2} |\vec{P}|^2 \right) - i \lambda P_j P_k \] The momenta are transformed under boosts in a nonlinear way we will discuss. Note that in the \(\lambda\to 0\) limit, you get the usual \(i\delta_{jk}P_0\).
The nonlinear function of the momentum generators on the right hand side looks arbitrary but it is not really arbitrary: the Hopf algebras have kind of constrained rules and this structure is forced upon you. Alternatively, you obtain very analogous nonlinear commutation relations when you study various important 2-dimensional solvable field theories that have "quantum groups" as their symmetries.
Note that if the energy and momenta \(P_0,P_i\) are conserved in one frame, they will be conserved in a frame boosted by the \(N_i\) generators as well. But the energy and momenta in the new frame will be a nonlinear function of the original ones. As I tried to explain more than five years ago, in Locality and additivity of energy, such a nonlinear formula for the total energy and the total momentum spoils locality.
Imagine that you have two subsystems in different parts of the Milky Way. Unless it is possible to send instantaneous signals, the two subsystems should evolve independently of one another. In the quantum setup, this means that the state describing both objects must approximately be equal to the tensor product of states describing each of them,
\[ |\psi\rangle = |\psi_1\rangle \otimes |\psi_2\rangle \] while the Hamiltonian acting on the overall state \(|\psi\rangle\) must be simply a sum
\[ H = H_1 + H_2 \] of the Hamiltonians acting on the subsystems. This is necessary for the independence of the systems (the tensor factorization of the overall state) to be preserved under time evolution. So the energy should be additive: each of the two systems should simply add the right term. For similar reasons, the total momentum must be the sum of the momenta of the distant pieces as well.
But if this additivity for the energy and momentum holds in one frame, it will inevitably be violated in another frame that you obtain by boosts generated by \(N_i\). Indeed, under these κ-boosts, the energy and momentum transform to
\[ (E,p_i)\to {\rm nonlinear\,\,gibberish} (E,p_i) \] which is too bad because nonlinear gibberish functions of the total \(E\) and the total \(p_i\) are obviously not sums of the pieces coming from the two places of the Milky Way.
There exist all kinds of mixings which prove that even if the locality of the laws of physics were valid in one reference frame, it won't hold in any other κ-deformed reference frame. The evolution of the object on our side of the Milky Way will strongly depend on all kinds of products of momenta of our object and the object on the other side of our Galaxy. That's too bad. One may show that different observers (who are boosted relatively to each other) will also inevitably disagree on whether or not an object is caught inside a box, and so on.
To summarize, decent laws of physics in more than 1+1 dimensions prohibit some brutal forms of nonlocal communication and respect some laws of locality, at least approximate ones. That's why theories with quantum-deformed symmetries are almost certainly physically uninteresting for our understanding of dynamics in spacetimes or world volumes above 1+1 dimensions.
And that's the memo.
Kappa-Poincaré algebra
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Reviewed by MCH
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October 06, 2011
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