Precausality and continuity in 1-postulate derivations of SR

[strangerep]

The mention of "commutativity" in the context of this thread arose from the fact that a 1-parameter Lie group is necessarily commutative. Here, it's the group of velocity boosts along an arbitrary, but fixed, direction.

In group-theoretic (1-postulate) derivations of SR, one can simplify some of the computations by using $B(v_1) B(v_2) = B(v_2) B(v_1)\,$, where $B(v)$ denotes a boost transformation along the chosen fixed direction with parameter $v$.

 

[jbergman]

The mention of "commutativity" in the context of this thread arose from the fact that a 1-parameter Lie group is necessarily commutative. Here, it's the group of velocity boosts along an arbitrary, but fixed, direction.

In group-theoretic (1-postulate) derivations of SR, one can simplify some of the computations by using $B(v_1) B(v_2) = B(v_2) B(v_1)\,$, where $B(v)$ denotes a boost transformation along the chosen fixed direction with parameter $v$.

Right. But I thought Enrico was implying that the transformation group for space time is $SO^{+}(1,3)$ because of this commutativity of the 1-parameter subgroups. I don't quite see the connection.

 

[strangerep]

Right. But I thought Enrico was implying that the transformation group for space time is $SO^{+}(1,3)$ because of this commutativity of the 1-parameter subgroups.

I don't think that's what Enrico was implying. (For a bit more context see posts #5, #6 in this thread.) But I should probably let Enrico speak for himself.

 

[Enrico]

I don't think that's what Enrico was implying. (For a bit more context see posts #5, #6 in this thread.) But I should probably let Enrico speak for himself.

As I said, I'm only considering 1 spatial dimension. I did not mean to extend the argument to 3D.

 

[jbergman]

As I said, I'm only considering 1 spatial dimension. I did not mean to extend the argument to 3D.

I finally read the Berzi and Gorini paper so have better context. In the last section of the paper they rule out the rotation group because of precausality.

The question of whether or not commutativity implies continuity seems a bit circular. Because we inferred commutativity from the fact that the image of a one-parameter group is commutative, which is a continuous maps on to a group of transformations.

If you throw out the fact that $f(v)$ is continuous then you have to prove commutativity directly from the principle of relativity, i.e., that $f(v_1)f(v_2)=f(v_2)f(v_1)$.

Continuity seems like a reasonable assumption to me, just as reasonable as homogeneity and isotropy of space. It just says that $f(v_1)$ is near $f(v_2)$ if $v_1$ is near $v_2$.

 

[Enrico]

The question of whether or not commutativity implies continuity seems a bit circular. Because we inferred commutativity from the fact that the image of a one-parameter group is commutative, which is a continuous maps on to a group of transformations.

If you throw out the fact that f(v) is continuous then you have to prove commutativity directly from the principle of relativity, i.e., that f(v1)f(v2)=f(v2)f(v1).

This is my understanding as well.

Continuity seems like a reasonable assumption to me, just as reasonable as homogeneity and isotropy of space. It just says that f(v1) is near f(v2) if v1 is near v2.

Of course it is reasonable. I'm just wondering whether it may be stripped down further. At the moment my opinion is that some continuity assumption is needed. First, in order to prove linearity, continuity at the origin is needed on the transformation formulae, at any fixed $v$ (please see the first part of the Berzi & Gorini paper). Second, it seems to me that continuity at the origin is necessary for the transformation coefficients, as functions of $v$. Maybe it is sufficient to impose continuity (at the origin) on $\gamma$ only, but I'm not sure of that yet.

In any case, I'm going to put the subject aside for the moment: I think it be proper for me to deepen my knowledge in group theory now, in order to be at least aware of the correct concepts and terminology. For what regards precausality, I'm still convinced that it is not needed in the end.

 

[vanhees71]

Indeed, only the subgroup that is continuously (or here in the case of Lie groups even smoothly) connected with the identity is necessarily a symmetry group for any closed system, if the physical laws should be consistent with the underlying spacetime structure. The "disconnected parts" can still be symmetries. In Nature that's only partially the case since the weak interaction violates parity and time reversal.

In the quantum realm, i.e., in local relativistic QFT you have in addition also charge conjugation, i.e., that for each particle there's also its antiparticle (in the case of strictly neutral ones both are the same). From the symmetry under $\mathrm{ISO}(1,3)^{\uparrow}$, i.e., the proper orthochronous Poincare group and locality (microcausality) it follows that CPT must also be a symmetry. The weak interaction fufills this symmetry since it's described by such a local QFT, but it breaks all the other "discrete" symmetries, i.e., P, T, CP, and CT. All this symmetry breakings are experimentally verified.