Precausality and continuity in 1-postulate derivations of SR

[Enrico]

[Moderator's note: Thread spun off from previous one due to topic shift.]

Please forgive my ignorance, I've never studied group theory systematically up to now, so I'm not aware of all the concepts and symbols that have been used up to now. Yet, I'm interested in the derivation of the Lorentz transformations, and I think I quite grasp the concept expressed in the last post (#16). A few years ago I made a bit of bibliographic research on the "1-postulate" derivations of the Lorentz transformations, and there are a couple of things I'm trying to clarify within that framework. These are related to the group structure of the set of the transformations.

Let me express my thoughts in a very humble way, without invoking group theory from the outset. Let's consider a single spatial dimension. So, we are dealing with linear transformations from $\mathbb{R}^2$ into itself, i.e. with $2\times2$ matrices. As far as I know, all existing "1-postulate" derivations assume isotropy of space and the group structure for the set of transformations (matrices), and also a couple of further "technical" things. The first is that the coefficients of the transformation (elements of the matrix) be (at least) continuous functions of the group parameter. The second is a "causality" or "precausality" assumption, stating that "if two events occur at the same place in some frame of reference, their time order must be the same in all other frames of reference" (or the equivalent condition that by composing two positive velocities one must obtain a positive velocity).

Let's focus first on the second of these two assumptions, the "precausality" condition. Is it really needed? From my present understanding, this condition is necessary if we impose group structure on $\mathcal{L}_+$. If we impose group structure on $\mathcal{L}_+^\uparrow$ instead, I think it is not necessary. This is related to the choice of the group parameter. If the latter is $v$, the relative velocity between the two frames of reference that we are transforming (which amounts to say that, given a frame, the other one is completely defined once $v$ be fixed), we are actually considering $\mathcal{L}_+^\uparrow$, so precausality is not needed.

Now, what about the other assumption, the one about continuity of the coefficients as functions of the group parameter ($v$)? I've been trying to do some calculations by hand (i.e. without any tool from group theory) in the course of the last week, and I have an idea that this assumption is not needed actually.

Just to get a feeling of the general opinion on these topics. Again, please forgive my present ignorance on group theory.

 

[vanhees71]

I think the assumption you call "precausality" (a pretty good wording, I think!) is absolutely necessary to establish any spacetime structure, because theoretical physics (usually tacitly) assumes causality, which restricts the symmetry group to $\mathrm{O}(1,3)^{\uparrow}$, i.e., the piece of the full Lorentz group, $\mathrm{O}(1,3)$ that keeps the time order of time-like events the same.

Usually one indeed assumes in addition that it is, however, the subgroup that is continuously connected with the identity, $\mathrm{SO}(1,3)$, that must be the symmetry group of special relativity as far as the consistency of physical (local, field-theoretical) models with the spacetime model is concerned, and indeed, Nature makes use of the freedom to violate symmetry under the full Lorentz group as well under the subgroup $\mathrm{SO}(1,3)$ through the weak interaction. So this additional assumption of continuity seems to be justified a posteriori.

 

[Enrico]

And what about the choice of $v$ as a parameter (in the case with one single spatial coordinate)? I mean, the assumption that, once $v$ be given, the transformation must be uniquely determined. May it be that this assumption automatically implies continuity and precausality, i.e. restricts the group to $SO^\uparrow\left(1,3\right)$?

 

[robphy]

Is $v$ a slope?
Given measures on the t and x-axis (before a quadratic form is somehow specified), the slope is defined…. It’s a pre-metrical concept.

 

[Enrico]

Is v a slope?
Given measures on the t and x-axis (before a quadratic form is somehow specified), the slope is defined…. It’s a pre-metrical concept.

As I mentioned in my first post, $v$ is the relative velocity between the two frames of reference that we are transforming, so yes, it's a slope.

I think (from some computations) that the continuity hypothesis may be dropped if we assume from the outset that the group be commutative. Does anybody have a justification for commutativity based on the Relativity Principle?

On the other hand, I'd also like to understand whether commutativity is really needed. I tried quite hard to do without it, but such an approach seems unlikely to work. At this point, would be nice to have a counterexample of a non-commutative group obeying the other assumptions of the 1-postulate derivations (and not satisfying continuity in $v$, necessarily).

 

[strangerep]

As I mentioned in my first post, $v$ is the relative velocity between the two frames of reference that we are transforming, so yes, it's a slope.

I think (from some computations) that the continuity hypothesis may be dropped if we assume from the outset that the group be commutative. Does anybody have a justification for commutativity based on the Relativity Principle?

I don't think it's even possible to have a 1-parameter nonabelian group. But mathematicians might say differently. Paging @fresh_42 ...

On the other hand, I'd also like to understand whether commutativity is really needed. I tried quite hard to do without it, but such an approach seems unlikely to work. At this point, would be nice to have a counterexample of a non-commutative group obeying the other assumptions of the 1-postulate derivations (and not satisfying continuity in $v$, necessarily).

A 1-parameter group means there is a map $\varphi : {\mathbb{R}} \to G$. (Here we of course assume that the parameter $v \in {\mathbb{R}}$.)

The definition of "continuity" for $\varphi$ is that any open subset $S$ of $G$ in $Range(\varphi)$ maps via $\varphi^{-1}$ to an open interval of ${\mathbb{R}}$.

If you want $\varphi$ not to be continuous, then you must find an open subset $S$ of $G$ such that $\varphi^{-1}$ does not map to an open interval on ${\mathbb{R}}$.

I don't think this makes much sense because if you abandon the property that $G$ is a differentiable manifold (which automatically comes with the standard Euclidean topology), then what is the topology on your $G$? Without specifying this topology, you cannot even talk about "continuity" in a mathematically sensible way.

 

[Enrico]

The definition of "continuity" for φ...

I'm talking about continuity of the coefficients of the transformation (elements of the matrix) as functions from $\mathbb{R}$ to $\mathbb{R}$, not of the group parameterization.

I don't think it's even possible to have a 1-parameter nonabelian group.

If that's the case, then the problem seems to be solved (continuity is not needed as an initial hypothesis).

 

[Rene Dekker]

I think the "pre-causality" follows from the "speed of light" postulate, and is therefore not a separate postulate. If light travels from event A to event B, then event A necessarily occurs before B. Through the "speed of light" postulate, that means that event A must occur before B in all reference frames.

For any two events A and C occurring at the same location in some reference frame, we can always imagine a light beam traveling from A to some other point B, and then back again, to arrive just in time for C. Therefore, A, B, and C must have the same causal relationship in all reference frames.

 

[vanhees71]

Indeed, the time-ordering between two events is invariant under orthochronous Lorentz transformations if the two events are time-like or light-like separated. To see this consider the four-vectors $x$ and $y$ and set $z=y-x$. Further assume that $z \cdot z \geq 0$ (that's what means timelike or like-like separated events, where I'm using the (1,3)-signature convention, i.e., $(\eta_{\mu \nu})=\mathrm{diag}(1,-1,-1,-1)$.

A orthochronous Lorentz transformation is a matrix which transforms from one Lorentzian basis to another, i.e., it's represented by a matrix ${\Lambda^{\mu}}_{\nu}$ fulfilling $\eta_{\rho \sigma} {\Lambda^{\rho}}_{\mu} {\Lambda^{\sigma}}_{\nu}=\eta_{\mu \nu}$, ${\Lambda^0}_0 \geq 1$.

Now consider two Lorentzian bases $e_{\mu}$ and $e_{\mu}'$ and assume that wrt. to the first one $x^0<y^0$, i.e., $z^0>0$. Then you have
$$z^{\prime \mu}={\Lambda^{\mu}}_{\nu} z^{\nu}$$
and
$$z^{\prime 0}={\Lambda^0}_0 z^0+{{\Lambda^0}_j} z^{j} \geq {\Lambda^0}_0 z^0 -|{\Lambda^0}_j z^j|.$$
For notational simplicity let's write ${\Lambda^0}_j z^j=\vec{\lambda} \cdot \vec{z}$. Then with the Cauchy-Schwarz inequality
$$z^{\prime 0} \geq {\Lambda^0}_0 z^0 -|\vec{\lambda} \cdot \vec{z}| \geq {\Lambda^0}_0 z^0 - |\vec{\lambda}| |\vec{z}|.$$
Now $({\Lambda^0}_0)^2-\vec{\lambda}^2 =1$ and thus because of ${\Lambda^0}_0 \geq 1$ we have
$${\Lambda^0}_{0} =\sqrt{1+\vec{\lambda}^2} > |\vec{\lambda}|.$$
Thus
$$z^{\prime 0} > |\vec{\lambda}| (z^0-|\vec{z}|).$$
Then finally by assumption $z \cdot z=(z^0)^2-\vec{z}^2 \geq 0$ and thus, because by assumption $z^0>0$, $z^0 \geq |\vec{z}|$, i.e.,
$$z^{\prime 0}>0.$$

 

[strangerep]

I think the "pre-causality" follows from the "speed of light" postulate, and is therefore not a separate postulate.

But this thread is about (so-called) "1-postulate" derivations of SR, hence does not assume the light postulate. So then the question is: "Does the light postulate follow from the pre-causality postulate?"

 

[robphy]

From my reading of the pre-causality condition,
it would hold true in Minkowski spacetime and the Galilean spacetime.

(In the Galilean case, it might be good to exclude infinite velocities… which I could call both “spacelike” (orthogonal to timelike) and “null” (zero norm) directions.
Imagine the opening up of the light cone.
That is to say, only “timelike” velocities are allowed.

In this case, it seems to me that one would also have a partial ordering as one has in Minkowski spacetime.)

 

[Enrico]

So then the question is: "Does the light postulate follow from the pre-causality postulate?"

For sure, a limit speed emerges, but its value is not determined (being possibly infinite). Yet, my question is: "Do we really need the pre-causality postulate?" I'm quite convinced that the answer is "no." At least as long as we assume the transformation to be uniquely determined by the value of the relative velocity $v$ between the two frames of reference under consideration.

Much more challenging is the problem related to the continuity hypothesis (in the sense discussed above). I'm not sure yet whether we can do without it; it seems to me that this is the case at least if we assume the group to be abelian (also, is it possible for this group not to be such? It has been said that a one-parameter group is always abelian, but is $v$ a parameter in the strict sense of Lie groups?).

 

[robphy]

Yet, my question is: "Do we really need the pre-causality postulate?" I'm quite convinced that the answer is "no." At least as long as we assume the transformation to be uniquely determined by the value of the relative velocity v between the two frames of reference under consideration.

I'm not sure if I followed all of the starting inputs to the question...
... but without the "pre-causality postulate" is the Euclidean rotation an allowed 2x2 matrix?

 

[strangerep]

Much more challenging is the problem related to the continuity hypothesis (in the sense discussed above). I'm not sure yet whether we can do without it; it seems to me that this is the case at least if we assume the group to be abelian (also, is it possible for this group not to be such? It has been said that a one-parameter group is always abelian, but is $v$ a parameter in the strict sense of Lie groups?).

Let's clear up a few mathematical details...

Your attempted distinction between group continuity $\varphi: \mathbb{R} \to G$ and continuity of the functions like $v \to f(v)$, i.e., a map from $\mathbb{R}$ to functions over $\mathbb{R}$ (which make up the components of Lorentz transformation matrices acting on spacetime vectors $V$) is an irrelevant distraction. Those matrices are a Representation of a Lie group. Such a representation is by definition a smooth homomorphism $\Pi: G \to GL(V)$. A homomorphism is a map between algebraic structures that preserves essential operations of those structures (in this case, multiplication, etc). Hence, one has continuity of the functions in the matrix representation (wrt to the real parameter) if and only if one has continuity with respect to the abstract group (wrt to the real parameter).

2) The term 1-parameter group usually means a continuous group homomorphism $\varphi: \mathbb{R} \to G$. So if you want a discontinuous relationship between relative velocity and the coordinate transformations corresponding to boosts of relative velocities, you'll have to choose some other mathematical structure. Without such a (mathematically precise) alternative structure, well,... sorry,... but you're just talking mathematical word salad nonsense.

 

[vanhees71]

I'm not sure if I followed all of the starting inputs to the question...
... but without the "pre-causality postulate" is the Euclidean rotation an allowed 2x2 matrix?

If you use the approach to the symmetry concepts of spacetime using only the special principle of relativity, i.e., the existence and indistinguishability of (global) inertial frames together with the assumption that space as observed by any inertial observer is Euclidean you end up with the Galilei, the Lorentz (Poincare) and the rotation group as possible spacetime symmetries, but the latter has to be excluded, because it does not obey a causality structure of spacetime in the sense that the temporal order of events is not independent under changes from one to another inertial frame, while this "causality assumption" is fulfilled for the Galilei group (which is pretty trivial, because Galilei-Newton spacetime leads to an absolute time) and the Poincare group. There time-like separated events have the same temporal order under changes from one frame of reference to another as long as you exclude the discrete time-reversal transformation, i.e., consider as the physical symmetry group of special-relativistic spacetime to be $\mathrm{SO}(1,3)^{\uparrow}$, which empirically also is the case since the weak interaction breaks space-reflection (parity) as well as time-reversal symmetry (as well as the extended but related symmetry under charge conjugation as well as CP and CT; it only respects CPT symmetry, as predicted by the assumption of local relativistic QFTs as the correct framework of relativistic QT).

Concerning the derivation of the Galilei and Lorentz transformations as symmetry groups of spacetimes fulfilling the special principle of relativity see, e.g.,

https://doi.org/10.1063/1.1665000

 

[Enrico]

Concerning the derivation of the Galilei and Lorentz transformations as symmetry groups of spacetime fulfilling the special principle of relativity see, e.g.,

https://doi.org/10.1063/1.1665000

This is exactly the main reference for my discussion. In that paper, the Authors assume explicitly continuity for the transformation parameters (as functions from $\mathbb{R}$ to $\mathbb{R}$, unless I'm missing anything they don't make any reference to Lie groups) and pre-causality (although this last assumption is not called exactly this way, again if I'm not wrong).

My opinion is that Euclidean rotations are not allowed even without assuming pre-causality, since they would require both positive and negative values for $\gamma(v)$ (for the same value of $v$). I may be very well wrong, but I can't see where, at the moment.

Those matrices are a Representation of a Lie group.

This would be a further assumption, or not? It's clear to me (I think...) that if we talk about Lie groups only, then continuity of the transformation parameters is equivalent to continuity of the group parametrization.

The term 1-parameter group usually means a continuous group homomorphism φ:R→G.

This is why I was asking whether $v$ be a parameter in the strict sense of Lie groups: if we want the parametrization to preserve algebraic operations, i.e.

$$ v_1 + v_2 \mapsto \varphi ( v_2 ) \circ \varphi ( v_1 ) $$

then $v$ is not a good parameter. Yet, assuming that $v$ uniquely identify the transformation seems reasonable.

 

[vanhees71]

The "natural" parameter for Lorentz boosts (in the sense of boosts in an arbitrary but fixed direction these form 1-parameter subgroups) is the rapidity rather than the "velocity". In 1+1D notation
$$L(\alpha)=\exp \left [\alpha \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix} \right]=\begin{pmatrix} \cosh \alpha & -\sinh \alpha \\ -\sinh \alpha & \cosh \alpha \end{pmatrix}.$$
Then $\beta=v/c=\tanh \alpha$, $\gamma=\cosh \alpha$.

 

[Enrico]

The "natural" parameter for Lorentz boosts (in the sense of boosts in an arbitrary but fixed direction these form 1-parameter subgroups) is the rapidity rather than the "velocity".

Yes. In the case $k<0$ (in the wording of Berzi & Gorini) one has Euclidean rotations, and the corresponding parameter would be the angle of rotation $\alpha$. My point is that in this case one has the same value of $v$ for two different values of $\alpha$ (within $[0,2/\pi[$). Do we allow that?

 

[vanhees71]

I think the main argument by Berzi&Gorini was that the rotation group of this putative spacetime model is indeed excluded by the "causality constraint", i.e., the restriction to transformations which leave temporal distances invariant leads to only a part of the rotation group, not forming a group.

 

[Enrico]

I think the main argument by Berzi&Gorini was that the rotation group of this putative spacetime model is indeed excluded by the "causality constraint", i.e., the restriction to transformations which leave temporal distances invariant leads to only a part of the rotation group, not forming a group.

Exactly. Yet, at the outset they assume the coefficients of the transformation to be uniquely identified by $v$ (see Eqs. (8) in their paper and the period right after Eqs. (10)). So why introducing a further constraint (causality)?

By the way, the causality constraint is not that temporal intervals be invariant, but that the temporal sequence of two given events be not inverted by the transformation (provided the two events have the same spatial coordinate in some frame of reference).

 

[PeterDonis]

the coefficients of the transformation (elements of the matrix) be (at least) continuous functions of the group parameter.

If this is the case, and the identity is a valid transformation, then all valid transformations must be continuously connected to the identity. That in itself is sufficient to restrict the transformations to the proper orthochronous ones, i.e., to $\mathrm{SO}(3,1)^{\uparrow}$.

 

[Enrico]

If this is the case, and the identity is a valid transformation, then all valid transformations must be continuously connected to the identity. That in itself is sufficient to restrict the transformations to the proper orthochronous ones, i.e., to $\mathrm{SO}(3,1)^{\uparrow}$.

I think my first post is misleading, since I used the symbols $\mathcal{L}_+$ and $\mathrm{SO}(3,1)$ from the outset, as if I were considering the Lorentz transformations as known and discussing their group topology. Instead, my aim is to discuss the derivation of the transformations.

In the case $k<0$, the full group of Euclidean rotations is continuously connected to the identity, right? So continuity would not be sufficient to restrict the transformations to the proper orthochronous ones, and as a consequence to exclude the case $k<0$ (in this case the proper orthochronous transformations constitute half of the group only).

 

[PeterDonis]

my aim is to discuss the derivation of the transformations.

Yes, I understand that. I was suggesting one argument for why the "precausality" condition is not necessary in such a derivation.

In the case $k<0$, the full group of Euclidean rotations is continuously connected to the identity, right?

Yes, so the continuity condition by itself would not exclude the $k < 0$ case. But if you have some other way of excluding that case without using the "precausality" condition (and you have suggested that requiring the transformation to be unique for a given value of $v$ would do that), then my suggestion would help to avoid the requirement for the "precausality" condition for the $k > 0$ case.

 

[vanhees71]

Exactly. Yet, at the outset they assume the coefficients of the transformation to be uniquely identified by $v$ (see Eqs. (8) in their paper and the period right after Eqs. (10)). So why introducing a further constraint (causality)?

By the way, the causality constraint is not that temporal intervals be invariant, but that the temporal sequence of two given events be not inverted by the transformation (provided the two events have the same spatial coordinate in some frame of reference).

Of course what I meant is that the temporal order is unchanged. Temporal intervals are of course not invariant in SR ("time dilation").

 

[robphy]

Let me express my thoughts in a very humble way, without invoking group theory from the outset. Let's consider a single spatial dimension. So, we are dealing with linear transformations from R2 into itself, i.e. with 2×2 matrices. As far as I know, all existing "1-postulate" derivations assume isotropy of space and the group structure for the set of transformations (matrices), and also a couple of further "technical" things. The first is that the coefficients of the transformation (elements of the matrix) be (at least) continuous functions of the group parameter. The second is a "causality" or "precausality" assumption, stating that "if two events occur at the same place in some frame of reference, their time order must be the same in all other frames of reference" (or the equivalent condition that by composing two positive velocities one must obtain a positive velocity).

Let's focus first on the second of these two assumptions, the "precausality" condition. Is it really needed?

Can you provide some specific references (from the various "1-postulate" derivations) for this "causality" or "precausality" assumption?
It would be helpful (for me) to see this formulated in a more precise in way
within the context of the associated structures and conditions assumed.

 

[strangerep]

Can you provide some specific references (from the various "1-postulate" derivations) for this "causality" or "precausality" assumption?
It would be helpful (for me) to see this formulated in a more precise in way
within the context of the associated structures and conditions assumed.

This is the Berzi + Gorini reference that Hendrik already quoted in post #15 above.

[For those without journal access, that paper seems to be downloadable here.]

 

[strangerep]

This is why I was asking whether $v$ be a parameter in the strict sense of Lie groups: if we want the parametrization to preserve algebraic operations, i.e. $$ v_1 + v_2 \mapsto \varphi ( v_2 ) \circ \varphi ( v_1 ) $$ then $v$ is not a good parameter. Yet, assuming that $v$ uniquely identify the transformation seems reasonable.

A Lie group parameter need not be strictly additive. It's quite ok to have the following:

For arbitrary $v_1, v_2$ in the parameter set $\mathbb{V}$, if $\varphi ( v_2 ) \circ \varphi ( v_1 ) \to \varphi ( v_3 )$, then $v_3 = f(v_1,v_2) \in \mathbb{V}$, where the function $f$ is a (continuous, differentiable) map $f : \mathbb{V}\times \mathbb{V} \to \mathbb{V}$.

 

[Enrico]

[...] that specific function is universal over the whole parameter set.

May you please specify formally the meaning of "universal" in this context? Is it referring to some universal property (I know some of the kind exists in the theory of categories and its neighbourhoods)?

Also, what about the condition that our group be a Lie one? My present ignorance on the subject prevents me from understanding how stringent it be and how it may be justified by initial (physical) assumptions.

 

[Enrico]

Some references:

• Lalan, M. V., Sur les postulats qui sont à la base des cinématiques. Bulletin de la Societe Mathematique de France 65, 83 (1937).
• Berzi, V. & Gorini, V., Reciprocity principle and the Lorentz transformations. Journal of Mathematical Physics 10, 1518 (1969).
• Lévy-Leblond, J.-M., One more derivation of the Lorentz transformation. American Journal of Physics 44, 271 (1976).
• Liberati, S., Sonego, S. & Visser, M., Faster-than-c signals, special relativity, and causality. Annals of Physics (N.Y.) 298, 167 (2002).
• Sonego, S. & Pin, M., Foundations of anisotropic relativistic mechanics. Journal of Mathematical Physics 50, 042902 (2009).

The paper by J.-M. Lévy-Leblond uses, in place of precausality, the condition that the composition of two positive velocities do not yield a negative velocity.

 

[strangerep]

May you please specify formally the meaning of "universal" in this context? [...]

Those extra words were probably extraneous, intended to help convey the meaning, but evidently not helping with that. I have edited down my post #27 to be (hopefully) more minimal and efficient.

Also, what about the condition that our group be a Lie one?

When constructing mathematical models in theoretical physics, one chooses the math framework(s) that seem best suited to the task. If, one day, some experimental fault or inconsistency emerges than shows Lie groups to be inadequate for current purposes, no doubt someone somewhere will think up something else.

My present ignorance on the subject prevents me from understanding how stringent it be and how it may be justified by initial (physical) assumptions.

This is a classic example where one should try very hard to become fully proficient in existing mainstream theory, (e.g., by working through recommended textbooks with pencil in hand) before trying to deconstruct it and build something else. Refusing to knuckle down and do this is one of the most reliable marks of a crackpot.

 

[Enrico]

When constructing mathematical models in theoretical physics, one chooses the math framework(s) that seem best suited to the task. If, one day, some experimental fault or inconsistency emerges than shows Lie groups to be inadequate for current purposes, no doubt someone somewhere will think up something else.

Of course. My aim is just to try to strip down the initial hypotheses as much as possible. That makes the theory more solid, right?

Refusing to knuckle down and do this is one of the most reliable marks of a crackpot.

I'm not refusing to do it, group theory is one of the topics I'd like to study systematically. I had started a few years ago, but life is made by many things, eventually I have devoted my free time to other hobbies. It's a matter of equilibrium, after the whole working day in an office it's not that easy to sit down and study maths. Anyway, the conversation here is a pleasant occasion to share my thoughts with others (rather than being isolated on my own) and get feedback, and I'm trying to get a feeling about topics commonly needed in mainstream theory and their relevance, before - hopefully - facing them systematically.

 

[jbergman]

As I mentioned in my first post, $v$ is the relative velocity between the two frames of reference that we are transforming, so yes, it's a slope.

I think (from some computations) that the continuity hypothesis may be dropped if we assume from the outset that the group be commutative. Does anybody have a justification for commutativity based on the Relativity Principle?

On the other hand, I'd also like to understand whether commutativity is really needed. I tried quite hard to do without it, but such an approach seems unlikely to work. At this point, would be nice to have a counterexample of a non-commutative group obeying the other assumptions of the 1-postulate derivations (and not satisfying continuity in $v$, necessarily).

I'm not sure what you mean by commutative in this context.

The restricted Lorentz group $SO^{+}(1,3)$ is a non-commutative group.

 

[jbergman]

And what about the choice of $v$ as a parameter (in the case with one single spatial coordinate)? I mean, the assumption that, once $v$ be given, the transformation must be uniquely determined. May it be that this assumption automatically implies continuity and precausality, i.e. restricts the group to $SO^\uparrow\left(1,3\right)$?

The single dimension case is much simpler than the 3 dimensional case. For instance, rotations in 1 and 2 dimensions, (SO(1), and SO(2)) form commutative groups but not in SO(3) for 3 dimensions does not.

 

[Enrico]

The single dimension case is much simpler than the 3 dimensional case. For instance, rotations in 1 and 2 dimensions, (SO(1), and SO(2)) form commutative groups but not in SO(3) for 3 dimensions does not.

Yes I was only considering the 1-dimensional case. I have mixed up symbols quite a bit, sorry. This is where my present lack of knowledge in group theory shows up clearly: I'm not sure what the symbol corresponding to $\mathrm{SO}^\uparrow(1,3)$ in one dimension would be.

 

[jbergman]

Yes I was only considering the 1-dimensional case. I have mixed up symbols quite a bit, sorry. This is where my present lack of knowledge in group theory shows up clearly: I'm not sure what the symbol corresponding to $\mathrm{SO}^\uparrow(1,3)$ in one dimension would be.

With one spatial dimension and one time dimension the that would be $\mathrm{SO}^\uparrow(1,1)$.

But, you can't draw conclusions about what happens in four dimensional space time from 2-dimensional space time because as I said commutativity in one case doesn't imply commutativity in the full space so I'm not sure how it helps your argument.