A Does QED Originate from Non-Relativistic Systems?

[A. Neumaier]

[Mentor's note - this thread was split off from https://www.physicsforums.com/threads/a-skeptics-view-on-bohmian-mechanics.899967/ as it is interesting in its own right and a digression there]

If we take the Wilsonian view of QFT (in the Copenhagen interpretation), then QED should be thought of as conceptually arising from a non-relativistic quantum mechanical system such as lattice QED.

This is a distortion of facts. One looks at lattice QCD only because it is derived from the Poincare invariant QCD; without the latter there would be no motivation at all to consider the former. And lattice QED is hardly ever pursued. The successes of QED, both historically and today, come solely from the Poincare invariant version.

 

[stevendaryl]

This is a distortion of facts. One looks at lattice QCD only because it is derived from the Poincare invariant QCD; without the latter there would be no motivation at all to consider the former. And lattice QED is hardly ever pursued. The successes of QED, both historically and today, come solely from the Poincare invariant version.

I agree with this. Lattice QED was specifically designed to have the right continuum limit. So it's not a good example if you're wanting to show that Lorentz invariance can arise natural as a continuum approximation to a non-invariant theory. To be convincing you would have to have an independent motivation for lattice QED that did not rely on having the right continuum limit.

 

[atyy]

I agree with this. Lattice QED was specifically designed to have the right continuum limit. So it's not a good example if you're wanting to show that Lorentz invariance can arise natural as a continuum approximation to a non-invariant theory. To be convincing you would have to have an independent motivation for lattice QED that did not rely on having the right continuum limit.

This is a distortion of facts. One looks at lattice QCD only because it is derived from the Poincare invariant QCD; without the latter there would be no motivation at all to consider the former. And lattice QED is hardly ever pursued. The successes of QED, both historically and today, come solely from the Poincare invariant version.

There is no Poincare invariant version!

 

[A. Neumaier]

There is no Poincare invariant version!

?

Any standard textbook of quantum field theory discusses Poincare invariant QED. The renormalized results at any loop order are Poincare invariant. And the experimental tests are about these results at low order (up to six).

That there are unresolved problems about the limit of suitably resummed nonperturbative version, which you seem to allude to, is a completely different matter that has at preesent no experimental consequences.

On the other hand, lattice QED is extremely inaccurate, very few computations have been performed, and if all other QED were erased, QED would immediately lose its status as an excellent physical theory.

 

[stevendaryl]

There is no Poincare invariant version!

Are you talking about no Poincare-invariant version of QED, or of lattice QED?

 

[atyy]

?

Any standard textbook of quantum field theory discusses Poincare invariant QED. The renormalized results at any loop order are Poincare invariant. And the experimental tests are about these results at low order (up to six).

That there are unresolved problems about the limit of suitably resummed nonperturbative version, which you seem to allude to, is a completely different matter that has at preesent no experimental consequences.

On the other hand, lattice QED is extremely inaccurate, very few computations have been performed, and if all other QED were erased, WED would immediately lose its status as an excellent physical theory.

Well, it is the resummed nonperturbative version that makes it a quantum theory. In the absence of that, there is no QED as a quantum theory. It is the lattice QED that is a proper quantum theory.

 

[atyy]

Are you talking about no Poincare-invariant version of QED, or of lattice QED?

The usual "Poincare invariant" QED has a landau pole, so it cannot exist as a quantum theory at all energies. If QED at high energies does not exist, the Poincare invariance is broken. So in effect, Poincare invariant QED is only a low energy effective field theory.

 

[A. Neumaier]

The usual "Poincare invariant" QED has a landau pole, so it cannot exist as a quantum theory at all energies.

The existence question for nonperturbative QED is wide open. The Landau pole has not been proved to exist, it may well be an artifact of low order perturbation theory. Your statement is therefore only a belief.

Poincare invariant QED is only a low energy effective field theory.

It is the latter that is Poincare invariant and responsible for all successes of QED.

Lattice theories are also only low energy effective field theories; so if you think the latter is a defect of a theory then lattice QED is as defective and far less predictive. There is no reason at all to give it the status you wish it to have.

 

[A. Neumaier]

it is the resummed nonperturbative version that makes it a quantum theory. In the absence of that, there is no QED as a quantum theory.

You may have this opinion but it is not shared by anybody else, as far as I can tell. There are many textbooks that have QED as one of the main examples of an excellent quantum theory, and none that says that there is no QED as a quantum theory.

 

[atyy]

You may have this opinion but it is not shared by anybody else, as far as I can tell. There are many textbooks that have QED as one of the main examples of an excellent quantum theory, and none that says that there is no QED as a quantum theory.

They all agree QED is only a low energy effective theory. It is only in this region that we need Poincare invariance. So we can think of lattice QED and Poincare invariant QED as the same in the sense that both give the same low energy effective theories. In other words, there is no need for true Poincare invariance, only effective.

 

[A. Neumaier]

They all agree QED is only a low energy effective theory.

Sure, but this is not a defect. All our theories in physics (with possible exception of string theory) are only low energy effective theories.

we can think of lattice QED and Poincare invariant QED as the same in the sense that both give the same low energy effective theories.

No we cannot. They don't give the same low energy effective theory. Everything is different about them.

Lattice QED in the form it exists makes not the same predictions but far weaker ones. Please cite a demonstration that lattice QED derives the anomalous magnetic moment to high accuracy!

In addition, lattice QED has one additional parameter, the lattice spacing, and all results depend on it. One gets the experimental results only in the limit where the lattice spacing goes to infinity and the coupling constants are highly tuned functions of the lattice spacing, and the fine tuning must be chosen exactly such that the results approach the QED limit - which presupposes it! The fine-tuning has no other justification!

 

[A. Neumaier]

@atyy: But all this on QED is off-topic here; if you want to continue, please open another thread, and we can go into details.

 

[atyy]

@atyy: But all this on QED is off-topic here; if you want to continue, please open another thread, and we can go into details.

The point is simple: my statements on lattice QED and QED are absolutely standard, although you may not like them, take for example: http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf, https://arxiv.org/abs/hep-lat/0211036.

There is of course fine tuning (without taking the lattice spacing to zero) so that the right low energy limit occurs, but all of this fine tuning is needed in the standard view of QED, and has nothing to do with Bohmian Mechanics.

Once one realizes that QED is non-relativistic in the standard view, there is no special problem for Bohmian Mechanics.

 

[atyy]

https://arxiv.org/abs/0808.0082

Can Lorentz invariance be maintained if there is an energy cutoff?

https://arxiv.org/abs/hep-lat/0211036 mentions other regularizations like dimensional regularization, but they are not gauge invariant. Also, it is not clear tha the other regularization construct a quantum theory. On the other hand, the lattice regularization does construct a quantum theory, from which Lorentz invariant QED can in principle emerge as a low energy effective theory.

http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf also mentions that one starts the construction of QFT with a lattice.

Certainly, asymptotic safety might still be possible. But till then, if we take the Wilsonian viewpoint, lattice QED gives us a secure conceptual starting point (although it is of course impractical for calculations).

 

[Demystifier]

So it's not a good example if you're wanting to show that Lorentz invariance can arise natural as a continuum approximation to a non-invariant theory.

A much better example is classical sound and corresponding quantum phonons. Sound satisfies a continuous Lorentz-invariant wave equation (with velocity of sound instead of velocity of light). Yet, it emerges from non-relativistic discrete theory of atoms. At the quantum level it illuminates particle creation/destruction in QFT, in the sense that creation and destruction of phonons really originates from processes in which no actual particles (atoms) are created or destructed.

 

[A. Neumaier]

dimensional regularization, but they are not gauge invariant.

dimensional regularization is gauge invariant. Indeed, this is one of the main reasons why it is used.

 

[A. Neumaier]

http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf also mentions that one starts the construction of QFT with a lattice.

and ends with a covariant theory. You take the start for the end. None of the references stops at the lattice. Their goal is always to get the covariant, physical theory, not the regularized one.

One gets the 10-decimal agreement of the anomalous magnetic moment only starting with the covariant version (and then making approximations, but not lattice approximations).

 

[A. Neumaier]

Lorentz invariant QED can in principle emerge as a low energy effective theory.

1-loop Lorentz invariant QED is fully (and higher loop QED conceptually) constructed without any cutoff or regularization or lattices in Scharf's book on quantum electrodynamics. And his book on a true ghost story does the same for other gauge theories.

 

[A. Neumaier]

A much better example is classical sound and corresponding quantum phonons. Sound satisfies a continuous Lorentz-invariant wave equation (with velocity of sound instead of velocity of light). Yet, it emerges from non-relativistic discrete theory of atoms.

Yes, but it is not a good example to argue that

QED is non-relativistic in the standard view

atyy has a very nonstandard understanding of the meaning of the word ''standard''. He means by it ''his personal standards'', not the standard of the current state of the art in physics.

 

[Demystifier]

atyy has a very nonstandard understanding of the meaning of the word ''standard''. He means by it ''his personal standards'', not the standard of the current state of the art in physics.

The viewpoint that @atyy advocates is not the standard viewpoint, but is a standard viewpoint.

 

[A. Neumaier]

The viewpoint that @atyy advocates is not the standard viewpoint, but is a standard viewpoint.

according to which standard? Anyone can define his own standard. But to deserve the word, a standard must be agreed upon. Is it perhaps the standard adopted by the Bohmian mechanics camp?

Whoever agrees to this standard has very low standards indeed.

 

[RockyMarciano]

The point is simple: my statements on lattice QED and QED are absolutely standard, although you may not like them, take for example: http://www.staff.science.uu.nl/~hooft101/lectures/basisqft.pdf, https://arxiv.org/abs/hep-lat/0211036.

There is of course fine tuning (without taking the lattice spacing to zero) so that the right low energy limit occurs, but all of this fine tuning is needed in the standard view of QED, and has nothing to do with Bohmian Mechanics.

Once one realizes that QED is non-relativistic in the standard view, there is no special problem for Bohmian Mechanics.

It makes little sense to claim that what is called by everyone relativistic QED is actually nonrelativistic QED, even people working on the latter make this distinction and knows what the difference between them is so clearly it can't be the standard view that relativistic QED is non-relativistic.
You may have the personal view that ultimately RQED is in some sense nonrelativistic and that may be debatable, but implying that lattice QED is the way to go is not very reasonable when its predictions are much poorer that RQED.
The Poincare invariance is obviously local (even if the hope is to prove that is rigourously global and therefore nonperturbative, but that hasn't been achieved so it is just a believe, ironically backed by results in the lattice), it is achieved order by order and term by term. And in practice that is all what's needed for precise predicitions to many decimal positions.
So I guess when you say it is not relativistic you refer to the order by order limitation, but that is not what is meant (mathematically in strict sense the restricted Lorentz group plus infinitesimal translations) when referring to QED as being Poincare invariant(even if in some textbooks and in some physicists minds both meanings of Poincare invariance, perturbative and nonperturbative, are conflated due to wishful thinking that assumes the latter as already achieved but simply lacking a formal proof as a minor detail.

 

[Demystifier]

according to which standard? Anyone can define his own standard. But to deserve the word, a standard must be agreed upon. Is it perhaps the standard adopted by the Bohmian mechanics camp?

Whoever agrees to this standard has very low standards indeed.

The Wilsonian viewpoint of renormalization is quite standard (and has nothing to do with BM camp).

 

[A. Neumaier]

The Wilsonian viewpoint of renormalization is quite standard (and gas nothing to do with BM camp).

Most effective field theories considered in particle physics are relativistic, and Wilson's renormalization group mediates between relativistic effective QFTs at various energies.

Invoking Wilson to justify an ideosyncratic ''standard'' is just a smokescreen.

 

[Demystifier]

Most effective field theories considered in particle physics are relativistic, and Wilson's renormalization group mediates between relativistic effective QFTs at various energies.

You have a too narrow view of Wilsonian RG, which is somewhat typical for high-energy physicists. By Wilsonian RG, the effective theory at large distances may have very different symmetries than the fundamental theory at small distances, which is related to the fact that "renormalizaton group" is not really a group. Generally, this point is much better understood in condensed-matter community than in high-energy community.

 

[A. Neumaier]

You have a too narrow view of Wilsonian RG, which is somewhat typical for high-energy physicists. By Wilsonian RG, the effective theory at large distances may have very different symmetries than the fundamental theory at small distances, which is related to the fact that "renormalizaton group" is not really a group. Generally, this point is much better understood in condensed-matter community than in high-energy community.

I didn't intend to summarize in one sentence the whole range of appications of Wildon's RG ideas.

Wilson's ideas have nothing to do with the distinction between relativistic and nonrelativistic. As I pointed out, they also apply to relativistic effective theories without any intermediate nonrelativistic intermediate, and hence cannot be used to justify that a relativistic theory is fundamentally nonrelativistic.

It can be used that a relativistic theory might be the effective theory of an underlying nonrelativistic theory, but whether the latter is actually the case is a completely different question. In fact, nobody has succeeded so far to derive QED as an effective theory of an underlying nonrelativistic theory, so whether this is possible is merely speculation.

Moreover, atyy had argued the opposite way, that the lattice approximation is already QED. This is not at all justified by Wilson's RG which loses degrees of freedom and only goes from an approximation to a coarser one. You can make a coarser lattice theory from a fine one by using Wilson's RG. But the latter is completely silent about the opposite direction, to get QED from its lattice approximations.

 

[Demystifier]

It can be used that a relativistic theory might be the effective theory of an underlying nonrelativistic theory, but whether the latter is actually the case is a completely different question.

Good point.

Moreover, atyy had argued the opposite way, that the lattice approximation is already QED.

Outside of the lattice community it looks wrong, but in the lattice community it may be a standard view.

 

[A. Neumaier]

in the lattice community it may be a standard view.

Why ''may be''? One can claim anything in the subjunctive, it means nothing.

I never have seen such a claim by those pursuing lattice QFT.
They never claim the results of covariant computations as successes of their lattice theories.

 

[RockyMarciano]

Outside of the lattice community it looks wrong, but in the lattice community it may be a standard view.

The lattice community distinguishes perfectly relativistic from nonrelativistic QED, and doesn't call the former nonrelativistic nor claims the latter to be the standard QED.

 

[Demystifier]

I never have seen such a claim by those pursuing lattice QFT.

Try
https://www.amazon.com/dp/0521599172/?tag=pfamazon01-20

 

[A. Neumaier]

Try
https://www.amazon.com/dp/0521599172/?tag=pfamazon01-20

From the short paragraph summarizing the content of the book on the page you linked to:

''Quantum fields exist in space and time, which can be approximated by a set of lattice points.''

This is the opposite of atyy's and your claim. They distinguish perfectly between approximations and the real thing, already in the advertisenment of their book.

 

[atyy]

1-loop Lorentz invariant QED is fully (and higher loop QED conceptually) constructed without any cutoff or regularization or lattices in Scharf's book on quantum electrodynamics. And his book on a true ghost story does the same for other gauge theories.

Scharf's construction is divergent, so it is not an example of Poincare invariant QED.

 

[A. Neumaier]

Scharf's construction is divergent,

There is no divergence in any of the Lorentz invariant 1-loop results provided by Scharf, and these give already much better agreement with experiment than lattice QED. 1-loop QED is the version of QED (though with a different derivation) for which Feynman, Tomonaga and Schwinger got the Nobel prize!

 

[atyy]

There is no divergence in any of the Lorentz invariant 1-loop results provided by Scharf, and these give already much better agreement with experiment than lattice QED. 1-loop QED is the version of QED (though with a different derivation) for which Feynman, Tomonaga and Schwinger got the Nobel prize!

Although each term is finite, the series is divergent.

 

[RockyMarciano]

Although each term is finite, the series is divergent.

That each term is finite is enogh to claim local Poincare invariance which is the actual mathematical claim of Poincare invariance in perturbative QED, the one that gives outstanding predictions.

 

[A. Neumaier]

Although each term is finite, the series is divergent.

This doesn't matter for prediction. The series is asymptotic, and the divergence is expected not to show up before loop order 100. The results at loop order 6 are already as accurate as the best experimental results.

Compare this with the poor accuracy obtained by noncovariant lattice theories. They give finite results at each lattice spacing, but none of the lattice spacings for which computations can be carried out gives results matching expoeriment, and it is not known whether (or in which sense) the limit exists in which the lattice spacing goes to zero. Thus lattice QED is according to your own criterion (convergence) in a worse state than the covariant theory.

 

[atyy]

That each term is finite is enogh to claim local Poincare invariance which is the actual mathematical claim of Poincare invariance in perturbative QED, the one that gives outstanding predictions.

This doesn't matter for prediction. The series is asymptotic, and the divergence is expected not to show up before loop order 100. The results at loop order 6 are already as accurate as the best experimental results.

Compare this with the poor accuracy obtained by noncovariant lattice theories. They give finite results at each lattice spacing, but none of the lattice spacings for which computations can be carried out gives results matching expoeriment, and it is not known whether (or in which sense) the limit exists in which the lattice spacing goes to zero. Thus lattice QED is according to your own criterion (convergence) in a worse state than the covariant theory.

If you only care about predictions, then QED is not a quantum theory!

What is the Hilbert space?

 

[Demystifier]

''Quantum fields exist in space and time, which can be approximated by a set of lattice points.''

Well, I don't remember where exactly I have seen the statement that lattice QFT can be viewed as a rigorous definition of QFT. But I know I have seen that, at more than one place.

 

[RockyMarciano]

If you only care about predictions, then QED is not a quantum theory!

What is the Hilbert space?

Quantum theory in its renormalized interacting quantum field form that gives accurate predictions is not depending on a Hilbert space in the usual concept of space of square integrable functions, but on more specific instances of Hilbert spaces like Hardy fixed point spaces.

 

[atyy]

Here is the state of the art:

At the physics level of rigour: we do not have a version of QED that extends to all energies, hence we do not have Poincare invariant QED. We do believe that QCD extends to all energies because of Asymptotic Freedom, hence we do have Poincare invariant QCD. However, gravity is not renornalizable, so we do not have a version of quantum gravity that is Poincare invariant. Certainly, these do not preclude future discoveries of Asymptotic Safety, or UV completions with more degrees of freedom introduced.

At the rigourous level, we do not have any 3+1 dimensional interacting Poincare invariant QFT.

 

[martinbn]

Just to be able to follow the subtle discussion, can some one explain what is meant by Poincare invariant QFT.

 

[RockyMarciano]

Here is the state of the art:

At the physics level of rigour: we do not have a version of QED that extends to all energies, hence we do not have Poincare invariant QED. We do believe that QCD extends to all energies because of Asymptotic Freedom, hence we do have Poincare invariant QCD. However, gravity is not renornalizable, so we do not have a version of quantum gravity that is Poincare invariant. Certainly, these do not preclude future discoveries of Asymptotic Safety, or UV completions with more degrees of freedom introduced.

At the rigourous level, we do not have any 3+1 dimensional interacting Poincare invariant QFT.

True, but this is not what is meant by the local, i.e. term by term Poincare invariance of QED, that is what many physicist hope for but hasn't been proved and is even subject of a million dollar prize.

 

[A. Neumaier]

What is the Hilbert space?

For practical purpuses (indeed, to make predictions with QED at finite times, e.g., to derive the macroscopic Maxwell equations) the Hilbert space is defined through the closed time path (CTP) formalism. Whether this has a rigorous formulation is an open problem.

If you only care about predictions, then QED is not a quantum theory!

Most of theoretical physics (including quantum mechanics) is not fully rigorous. Quantum mechanics is not a sub-discipline of mathematical physics; only the latter requires everything to be rigorously defined.

 

[vanhees71]

A Poincare invariant QFT is one leading to a Poincare invariant S matrix. There is no rigorous definition of a nonperturbative/exact interacting QED in 1+3 spacetime dimensions. We only have the formalism for loop-order-by-loop-order covariant perturbation theory, and for QED that's quite satisfactory in comparison to experiment (Lamb shift of the hydrogen atom, anomalous magnetic moment of the electron, etc.).

Further, it's clear that to define the perturbative expressions beyond tree level you have to either first regularize and then renormalize (the standard way since 1971 is to use dimensional regularization, because it's most convenient by satisfying the important symmetries like gauge and Poincare symmetries for a large class of models) or to just renormalize by using the BPHZ formalism which subtracts systematically all divergences of the loop integrals directly in the integrands. This is, however just a technical question. The final result are the S-matrix elements expressed in renormalized and finite quantities at any order of perturbation theory.

The lattice approach is a special way to regularize QFT, and it's used as a non-perturbative approach (mostly in QCD, not QED). Also there of course, the final results one is really interested in are the continuum extrapolations, as Arnold stressed several times.

Of course, there's also a non-relativistic limit of both QED and QCD, but I don't see, in how far this is related to this quite general discusssion.

 

[Demystifier]

Just to be able to follow the subtle discussion, can some one explain what is meant by Poincare invariant QFT.

I think you brilliant answer can be applied:
An argument against Bohmian mechanics?


Now seriously, it is QFT whose action is invariant under Poincare group, where Poincare group is
https://en.wikipedia.org/wiki/Poincaré_group
See also Sec. 1.5 of Ticciati, QFT for Mathematicians (I think you said you liked that book).

 

[atyy]

Most of theoretical physics (including quantum mechanics) is not fully rigorous. Quantum mechanics is not a sub-discipline of mathematical physics; only the latter requires everything to be rigorously defined.

Sure, but at the non-rigourous level it is widely agreed that QCD, but not QED, has been demonstrated to exist at all energies.

 

[A. Neumaier]

What is the Hilbert space?[/QUOTE]

lattice QFT can be viewed as a rigorous definition of QFT.

It can be viewed as a rigorous definition of approximations only. In the view of theoretical physics this may be rigorously enough, but renormalized perturbation theory (with selective resummation in case of strong coupling) is as rigorous as lattice theory.

 

[A. Neumaier]

At the physics level of rigour: we do not have a version of QED that extends to all energies, hence we do not have Poincare invariant QED. We do believe that QCD extends to all energies because of Asymptotic Freedom, hence we do have Poincare invariant QCD.

This is a strange view of the state of the art.

We have a covariant version of QED valid to any loop order, giving excellent computational results at all energies that will ever be experimentally accessible, far beyond the Planck level. We also have a Poincare invariant QCD, not because of a belief that only affects energies beyond all experimental accessibility but because of renormalization group improved perturbation theory (needed because of the strong interaction), though it is still in a far less good state than QED.

 

[A. Neumaier]

Sure, but at the non-rigourous level it is widely agreed that QCD, but not QED, has been demonstrated to exist at all energies.

It is just a matter of belief, not of demonstration. Whereas it is a well-known fact that QED is Poincare invariant in each finite loop approximation. and is demonstated to have a predictive power far beyond QCD. What happens at energies above the Planck scale is completely irrelevant for QED; it is a matter for the future unified theory.

Perturbative QED does not suffer from the Landau pole; only a nonperturbative version possibly does, if one attempts to construct the theory using lattices (!) or using a cutoff. The Landau pole invalidates a construction only if the (lattice or energy) cutoff has to move through the pole in order to provide a covariant limit.

On the other hand, while the causal, covariant construction of QED also has a Landau pole, Landau's argument that a Landau pole invalidates perturbation theory no longer applies to this version. In the causal construction there are no cutoffs that must be sent to zero or infinity and move through the pole. The pole is only in the choice of the renormalization point (or mass $M$). But in the exact theory, the theory is completely independent of this renormalization point; so it can be chosen at low energy without invalidating the construction!

The associated Callan-Symanzik equation shows how the observables depend on the chosen renormalization mass $M$, giving an identical theory - apart from truncation errors, which are of course small only if $M$ is of the order of the energies at which predictions are made. Thus for any range of $M$ for which the Callan-Symanzik equation is solvable, one gets the same theory. The Landau pole found (nonrigorously) in causal QED in the Callan-Symanzik equation only means that one cannot connect the theory defined by a superhigh renormalization point irrelevant for the physics of the universe to the theory defined by a renormalization point below the Landau pole. This is a harmless situation.

It is quite different from Landau's argument that the cutoff cannot be removed because on the way from a small cutoff energy to an infinite cutoff energy perturbation theory becomes invalid since the terms become infinite when the cutoff passes the pole. This means that perturbation theory fails close to this cutoff. Thus any approximate construction with cutoff featuring a Landau pole cannot be made to approximate the covariant QED to arbitrary precision. In particular, the Landau pole believed to exist in lattice QED is of this fatal kind and proves (at this level of rigor) that lattice QED can never approach the covariant QED, hence is a fake theory. Your insistence on the Landau pole defeats even your basic goal!

 

[A. Neumaier]

Just to be able to follow the subtle discussion, can some one explain what is meant by Poincare invariant QFT.

Poincare invariant = Lorentz invariant and translation invariant.

Translation invariance guarantees the conservation of energy and momentum, and Lorentz invariance is the minimal requirement for relativistic causality. (It is not quite sufficient since for causality one also needs hyperbolicity.)