# Does Bell's theorem imply other Lorentz transformations?

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## Main Question or Discussion Point

Could it be that the transformations keeping the wave equation invariant have other solutions than the usual Lorentz ones ?

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PeterDonis
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Could it be that the transformations keeping the wave equation invariant have other solutions than the usual Lorentz ones ?
AFAIK the mathematical equivalence of "keep the wave equation invariant" and "Lorentz transformation" has been proven, so I would say no.

What does this have to do with Bell's Theorem?

It is the conclusion of John Bell to this contradiction in his original paper On the Einstein-Podolsky-Rosen Paradox.

PeterDonis
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It is the conclusion of John Bell to this contradiction in his original paper On the Einstein-Podolsky-Rosen Paradox.
What conclusion are you talking about? Bell doesn't talk about the wave equation at all in this paper.

It is the conclusion of John Bell to this contradiction in his original paper On the Einstein-Podolsky-Rosen Paradox.

Last edited:
king vitamin
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There might be a perceived issue between non-local quantum correlations and Lorentz invariance, but there actually is not. In a Lorentz-invariant quantum theory, one demands that for any two local observables, $A(x,t)$ and $B(x',t')$, their commutator must vanish outside of each other's light cone:
$$[A(x,t),B(x',t')] = 0, \qquad (x - x')^2 > (t - t')^2.$$
This does not mean that you cannot have superluminal correlations. In general, one has
$$\langle A(x,t)B(x',t')\rangle \neq 0$$
even for $(x - x')^2 > (t - t')^2$. These are the correlations which Bell's theorem concerns itself with.

The reason this protects causality is that the vanishing of the commutator immediately implies that the causal order in which $A$ and $B$ are measured cannot matter. Alice can measure before Bob, or Bob can measure before Alice, and the results are necessarily correlated in the same way (and in such a way that the order of measurements cannot be determined). Therefore Alice and Bob cannot signal to each other using this method, since the physics cannot distinguish who measured first.

Addendum : the OP should read Bell's theorems in the sense :

1) Bell's inequality violation implies there cannot be a common cause in the past light cone

2) Leggett-Garg inequality violation implies nonlocality cannot explain quantum correlations

Hence there should be more elements of physical reality in the Lorentz transformations ?

Nugatory
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Hence there should be more elements of physical reality in the Lorentz transformations ?
Now we have to ask you what the Lorentz transformations have to do with past light cones?

For example Since the Lorentz' gamma factor is : $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ nothing can move faster than light, at least when avoiding imaginary numbers.

In fact with the speed of light moving entities are not possible neither (as observers at least ?).

The latter point could indicate that there lacks something in the derivation ?

Nugatory
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None of this is making any sense. The Lorentz transformations are coordinate transformations, a mathematical rule for calculating the coordinates assigned by one coordinate system to a point in spacetime from the coordinates assigned by another coordinate system. The physics is the same no matter what coordinates we use and how we transform them.

The infinities and complex numbers that appear when we set $v\ge c$ in the Lorentz transformations do not indicate any problem with the derivation. They’re just telling us that it is not possible to construct two coordinate systems that: are inertial (the coordinate acceleration of an object not subject to proper acceleration is zero); the coordinate speed of light is the same in both; and the relative speed between their spatial origins is greater than or equal to $c$.

Ibix
The latter point could indicate that there lacks something in the derivation ?
The second postulate of SR says that the speed of light is the same in all inertial reference frames. Thus an inertial reference frame (the mathematical formalisation of what we mean by "an observer") moving at the speed of light is self-contradictory - light would have to be both stationary and moving at 3×108m/s. It's not a problem with the derivation that won't let you describe observers moving at the speed of light - it's a fundamental tenet of the theory!

Things are perfectly free to move at the speed of light (as long as they're massless, it turns out). They just can't have reference frames attached to them.

PeroK
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For example Since the Lorentz' gamma factor is : $\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$ nothing can move faster than light, at least when avoiding imaginary numbers.

In fact with the speed of light moving entities are not possible neither (as observers at least ?).

The latter point could indicate that there lacks something in the derivation ?
What you're really trying to say, I imagine, is: why does the failure of Bell's theorem not cast doubt on SR, and in particular $c$ as the maximum speed at which information can be transmitted, and with that the validity of the Lorentz Transformation?

My question is wether the gamma factor were not a cross section of an infinite peak but that in a section a bit apart from this in another dimension there were in fact no singularity ?

PeterDonis
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My question is wether the gamma factor were not a cross section of an infinite peak but that in a section a bit apart from this in another dimension there were in fact no singularity ?
I have no idea what you mean by this. Nor do I see what it has to do with the thread topic.