## Saturday, December 31, 2005 ... //

### Testing E=mc2 for centuries

Chad Orzel seems to disagree with my comments about the interplay between the theory and experiment in physics. That's too bad because I am convinced that a person who has at least a rudimentary knowledge about the meaning, the purpose, and the inner workings of physics should not find anything controversial in my text at all.

Orzel's text is titled "Why I could never be a string theorist" but it could also be named "Why I could never be a theorist or something else that requires to use the brain for extended periods of time". Note that the apparently controversial theory won't be string theory; it will be special relativity. The critics who can't swallow string theory always have other, much older and well-established theories that they can't swallow either.

The previous text about the theory vs. experiment relations

Recall that I was explaining a trivial fact that in science in general and physics in particular, we can predict the results of zillions of experiments without actually doing them. It's because we know general enough theories - that have been found by combining the results of the experiments in the past with a great deal of theoretical reasoning - and we know their range of validity and their accuracy. And a doable experiment of a particular kind usually fits into a class of experiments whose results are trivially known and included in these theories. This is what we mean by saying that these are the correct theories for a given class of phenomena. An experiment with a generic design is extremely unlikely to be able to push the boundaries of our knowledge.

When we want to find completely new effects in various fields, we must be either pretty smart (and lucky) or we must have very powerful apparata. For example, in high-energy physics, it's necessary that we either construct accelerators that accelerate particles to high energies above 100 GeV or so - this is why we call the field high-energy physics - or we must look for some very weak new forces, for example modifications of gravity at submillimeter experiments, or new, very weakly interacting particles. (Or some new subtle observations in our telescopes.)

If someone found a different, cheaper way to reveal new physics, that would be incredible; but it would be completely foolish to expect new physics to be discovered in a generic cheap experiment.

Random experiments don't teach us anything

It's all but guaranteed that if we construct a new low-energy experiment with the same particles that have been observed in thousands of other experiments and described by shockingly successful theories, we are extremely unlikely to learn anything new. This is wasting of taxpayers' money especially if the experiments are very expensive.

In the particular case of the recent "E=mc^2 tests", the accuracy was "10^{-7}" while we know experimentally that the relativistic relations are accurate with the accuracy "10^{-10}", see Alan Kostelecky's website for more concrete details. We just know that we can't observe new physics by this experiment.

Good vs. less good experiments

In other fields of experimental physics, there are other rules - but it is still true that one must design a smart enough experiment to be able to see something new or to be able to measure various things (or confirm the known physical laws) with a better accuracy than the previous physicists. There are good experimentalists and less-good experimentalists (and interesting and not-so-interesting experiments) which is the basic hidden observation of mine that apparently drives Ms. or Mr. Orzel up the wall.

Once again: What I am saying here is not just a theorist's attitude. Of course that it is also the attitude of all good experimentalists. It is very important for an experimentalist to choose the right doable experiments where something interesting and/or new may be discovered (or invented) with a nonzero probabilitity. There is still a very large difference between the experiments that reveal interesting results or inspire new ideas and experiments that no one else finds interesting.

Every good experimentalist would subscribe to the main thesis that experiments may be more or less useful, believe me. Then there are experimentalists without adjectives who want to be worshipped just for being experimentalists and who disagree with my comments; you may guess what is the reason.

Of course that one may design hundreds of experiments that are just stamp-collecting - or solving a homework problem for your experimental course. I am extremely far from thinking that this is the case everywhere outside high-energy physics. There have been hundreds of absolutely fabulous experiments done in all branches of physics and dozens of such experiments are performed every week. But there have also been thousands of rather useless experiments done in all these fields. Too bad if Ms. or Mr. Orzel finds it irritating - but it is definitely not true that all experiments are created equal.

Interpreting the results

Another issue is that if something unexpected occured in the experiment that was "testing E=mc^2", the interpretation would have to be completely different than the statement that "E=mc^2" has been falsified. It is a crackpot idea to imagine that one invents something - or does an experiment with an iron nucleus or a bowl of soup - that will show that Einstein was stupid and his very basic principles and insights are completely wrong.

Hypothetical deviations from the Lorentz invariance are described by terms in our effective theories. Every good experimentalist first tries to figure out which of them she really measures. Neither of these potential deviations deserves the name "modification of the mass-energy relation" because even the Lorentz-breaking theories respect the fact that since 1905, we know that there only exists one conserved quantity to talk about - mass/energy - that can have various forms. We will never return to the previous situation in which the mass and energy were thought to be independent. It's just not possible. We know that one can transform energy into particles and vice versa. We can never unlearn this insight.

New physics vs. crackpots' battles against Einstein

Einstein was not so stupid and the principles of his theories have been well-tested. (The two parts of the previous sentence are not equivalent but they are positively correlated.) To go beyond Einstein means to know where is the room for any improvement, clarification, or deformation of his theories and for new physics, and the room is simply not in the space of ideas that "E=mc^2 is wrong" or "relativity is flawed". A good experimentalist must know something about the theory, to avoid testing his own laymen's preconceptions about physics that have nothing to do with the currently open questions in physics.

Whether or not an experimental physicist likes it or not, we know certain facts about the possible and impossible extensions and variations of the current theories - and a new law that "E=mc^2" will be suddenly violated by one part in ten million in a specific experiment with a nucleus is simply not the kind of modification that can be done with the physical laws as we know them. Anyone who has learned the current status of physics knows that this is not how serious 21st century physics looks like. The current science is not about disproving some dogmatic interpretations of Bohr's complementarity principle either.

Chad Orzel is not the only one who completely misunderstands these basic facts. Hektor Bim writes:

• Yeah, this post from Lubos blew me away, and I’ve been trained as a theorist.

Well, it does not look like a too well-trained one.

• As long as we are still doing physics (and not mathematics), experiment rules.

Experiments may rule, but there are still reasonable (and even exciting) experiments and useless (and even stupid) experiments. If someone thinks that the "leading role" of the experiments means that the experimentalists' often incoherent ideas about physics are gonna replace the existing theories of physics and that every experiment will be applauded even if it is silly, is profoundly confused. Weak ideas will remain weak ideas regardless of the "leading role" of the experiments.

• What also blew me away is that Lubos said that “There is just no way how we could design a theory in which the results will be different.” This is frankly incredible. There are an infinite number of ways that we could design the theory to take into account that the results would be different.

Once again, there are no ways how to design a scientific theory that agrees with the other known experiments but that would predict a different result of this particular experiment. If you have a theory that agrees with the experiments in the accelerators but gives completely new physics for the iron nucleus, you may try to publish it - but don't be surprised if you're described as a cook.

Of course that crackpots always see millions - and the most spectacular among them infinitely many ;-) - ways to construct their theories. The more ignorant they are about the workings of Nature, the more ways to construct the theories of the real world they see. The most sane ones only think that it is easy to construct a quantum theory of gravity using the first idea that comes to your mind; the least sane ones work on their perpetuum mobile machines.

I only mentioned those whose irrationality may be found on the real axis. If we also included the cardinal numbers as a possible value of irrationality, a discussion of postmodern lit crits would be necessary.

Scientific theories vs. crackpots' fantasies

Of course someone could construct a "theory" in which relativity including "E=mc^2" is broken whenever the iron nuclei are observed in the state of Massachusetts - much like we can construct a "theory" in which the law of gravity is revoked whenever Jesus Christ is walking on the ocean. But these are not scientific theories. They're unjustifiable stupidities.

The interaction between the theory and experiments goes in both ways

It is extremely important for an experimental physicist to have a general education as well as feedback from the theorists to choose the right (and nontrivial) things to measure and to know what to expect. It is exactly as important as it is for a theorist to know the results of the relevant experiments.

Another anonymous poster writes:

• What Lumo seems to argue is that somehow we can figure out world just by thinking about it. This is an extremely arrogant and short-sighted point of view, IMPO – and is precisely what got early 20th century philosophers in trouble.

What I argue is that it is completely necessary for us to be thinking about the world when we construct our explanations of the real world as well as whenever we design our experiments. And thinking itself is responsible at least for one half of the big breakthroughs in the history of science. For example, Einstein had deduced both special relativity as well as general relativity more or less by pure thought, using only very general and rudimentary features of Nature known partially from the experiments - but much more deeply and reliably from the previous theories themselves. (We will discuss Einstein below.)

Thinking is what the life of a theoretical physicist is mostly about - and this fact holds not only for theoretical physicists but also other professions including many seemingly non-theoretical ones. If an undereducated person finds this fact about the real world "arrogant", it is his personal psychological problem that does not change the fact that thinking and logical consistency are among the values that matter most whenever physical theories of the real world are deduced and constructed.

The anonymous poster continues:

• By the same reasoning the orbits of the planets must be circular – which is what early “philosophers” argued at some point.

Circular orbits were an extremely useful approximation to start to develop astrophysics. We have gone through many other approximations and improvements, and we have also learned how to figure out which approximations may be modified and which cannot. Cutting-edge physics today studies neither circular orbits nor the questions whether "E=mc^2" is wrong; it studies very different questions because we know the answers to the questions I mentioned.

Pure thought in the past and present

A wise physicist in 2005 respects the early scientists and philosophers for what they have done in the cultural context that was less scientifically clear than the present era, but she clearly realizes their limitations and knows much more than those early philosophers. On the other hand, a bad and arrogant scientist in 2005 humiliates the heroes of the ancient science although he is much more dumb than they were, and he is asking much more stupid questions and promoting a much more rationally unjustifiable criticism of science in general than the comparably naive early philosophers could have dreamed about.

Of course that in principle, one can get extremely far by pure thought, if the thought is logically coherent and based on the right principles, and many great people in the history of science indeed had gotten very far. These are the guys whom we try to follow, and the fact that there have been people who got nowhere by thinking cannot change the general strategy either.

• Anthropic principle completely destroys whatever is left of the “elegance” argument, which is why it’s entertaining to see what will happen next.

I know that some anti-scientific activists would like to destroy not only the "elegance" of science but the whole science - and join forces with the anthropic principle or anything else if necessary - but that does not yet mean that their struggle has any chance to succeed or that we should dedicate them more than this single paragraph.

Another anonymous user writes:

• As far as what Lubos meant, only he can answer that. But it would be obviously foolish to claim relativity could have been deduced without experimental input, and Lubos, whatever else he might be, is no fool.

History of relativity as a victory of pure thought

If interpreted properly, it would not be foolish; it is a historical fact. For example, I recommend you The Elegant Universe by Brian Greene, Chapter 2, for a basic description of the situation. Einstein only needed a very elementary input from the experiments - namely the invariance of physical laws under uniform motion; and the constancy of speed of light - which naturally follows from Maxwell's equations and Einstein was sure that the constancy was right long before the experiments showed that the aether wind did not exist.

It is known pretty well that the Michelson-Morley experiments played a rather small role for Einstein, and for some time, it was even disputed whether Einstein knew these experiments at all back in 1905. (Yes, he did.) Some historians argue that the patented ideas about the train synchronization themselves played a more crucial role. I don't believe this either - but the small influence of the aether wind experiments on Einstein's thinking seems to be a consensus of the historians of science.

Einstein had deeply theoretical reasons to be convinced about both of these two assumptions. Symmetry such as the Galilean/Lorentz symmetry or "the unity of physical explanations" are not just about some irrelevant or subjective concepts of "beauty". They are criteria that a good physicist knows how to use when he or she looks for better theories. The observation that the world is based on more concise and unified principles than what the crackpots and laymen would generally expect is an experimentally verified fact.

These two observations are called the postulates of special relativity, and the whole structure of special relativity with all of its far-reaching consequences such as the equivalence of matter and energy follows logically. Needless to say, all of these effects have always been confirmed - with accuracy that currently exceeds the accuracy available to the experimentalists of Einstein's era by very many orders of magnitude. Special relativity is a genuine and true constraint on any theory describing non-gravitational phenomena in our Universe, and it is a strong constraint, indeed.

Importance of relativity

Whoever thinks that it is not too important and a new experiment with a low-energy nucleus may easily show that these principles are wrong, which essentially allows us to ignore special relativity, and that everything goes after all, is a crackpot.

General relativity: even purer thought

In a similar way, the whole structure of general relativity was derived by the same Einstein purely by knowing the previous special theory of relativity plus Newton's approximate law of gravity, including the equivalence of the inertial and gravitational mass; the latter laws were 250 years old. There was essentially no room for experiments. The first experiments came years after GR was finished, and they always confirmed Einstein's predictions.

The known precession of Mercury's perihelion is an exception; this prediction of GR was known before Einstein, but Einstein only calculated the precession after he had completed his GR, and henceforth, the precession could not directly influence his construction of GR. He was much more influenced and impressed by Ernst Mach, an Austrian philosopher. I don't intend to promote Mach - but my point definitely is to show that the contemporary experiments played a very small role when both theories of relativity were being developed.

There were also some experiments that argued that they rejected the theory, and Einstein knew that these experiments had to be wrong because "God was subtle but not malicious". Of course that Einstein was right and the experiments were wrong. (Similar stories happened to many great theoretical physicists; an experiment of renowned experimentalists that claimed to have falsified Feynman-Gell-Mann's theory of V-A interactions was another example - and Feynman knew right away when he was reading the paper that the experimentalists were just being silly.) Our certainty today that special relativity (or the V-A nature of the weak interactions) is correct in the "simply doable" experiments is much higher than our confidence in any single particular experimentalist. You may be sad or irritated, but that's about everything that you can do against this fact.

Other theories needed more experiments

It would be much harder to get that far without experiments in quantum mechanics and particle physics, among many other branches of physics and science, but whoever questions the fact that there are extremely important insights and principles that have been found - and/or could be found or can be found - by "pure thought" (or that were correctly predicted long before they were observed), is simply missing some basic knowledge about science.

Although I happily admit that we could not have gotten that far without many skillful (and lucky) experimentalists and their experiments, there have been many other examples beyond relativity in which important theories and frameworks were developed by pure mathematical thinking whose details were independent of experiments. The list includes, among hundreds of other examples,

• Dirac's equation. Dirac had to reconcile the first-order Schrödinger equation with special relativity. As a by-product, he also predicted something completely unknown to the experimentalists, namely antiparticles. Every successful prediction may be counted as an example of theoretical work that was not driven by experiments.
• Feynman's diagrams and path integral. No one ever really observed "diagrams" or "multiple trajectories simultaneously contributing to an experiment". Feynman appreciated Dirac's theoretical argument that the classical concept of the action (and the Lagrangian) should play a role in quantum mechanics, too, and he logically deduced that it must play role because of his sum over trajectories. The whole Feynman diagram calculus for QED (generalizable to all other QFTs) followed by pure thought. Today we often say that an experiment "observes" a Feynman diagram but you should not forget about the huge amount of pure thought that was necessary for such a sentence to make any sense.
• Supersymmetry and string theory. I won't provoke the readers with a description.

Lorentz violations are not too interesting and they probably don't exist

• If he is claiming that Lorentz invariance must be exact at all scales, then I agree that he’s being ridiculous. But I think it is reasonable to claim that this experiment was not really testing Lorentz invariance at a level where it has not been tested before.

What I am saying is that it is a misguided approach to science to think that the next big goal of physics is to find deviations from the Lorentz invariance. We won't find any deviations. Most likely, there aren't any. The hypotheses about them are not too interesting. They are not justified. They don't solve any puzzles. Even if we find the deviations and write down the corresponding corrections to our actions, we will probably not be able to deduce any deep idea from these effects. Since 1905 (or maybe the 17th century), we know that the Lorentz symmetry is as fundamental, important and natural as the rotational symmetry.

The Lorentz violation is just one of many hypothetical phenomenological possibilities that can in principle be observed, but that will probably never be observed. I find it entertaining that those folks criticize me for underestimating the value of the experiments when I declare that the Lorentz symmetry is a fundamental property of the Universe that holds whenever the space is sufficiently flat. Why is it entertaining? Because my statement is supported by millions of accurate experiments while their speculation is supported by 0.0001 of a sh*t. It looks like someone is counting negative experiments as evidence that more such experiments are needed.

The only reason why the Lorentz symmetry irritates so many more people than the rotational symmetry is that these people misunderstand 20th century physics. From a more enlightened perspective, the search for the Lorentz breaking is equally (un)justified as a search for the violation of the rotational symmetry. The latter has virtually no support because people find the rotational symmetry "natural" - but this difference between rotations and boosts is completely irrational as we have known since 1905.

Parameterizing Lorentz violation

In the context of gravity, the deviations from the Lorentz symmetry that can exist can be described as spontaneous symmetry breaking, and they always include considering the effect of gravity as in general relativity and/or the presence of matter in the background. In the non-gravitational context, these violations may be described by various effective Lorentz-breaking terms, and all of their coefficients are known to be zero with a high and ever growing degree of accuracy. Look at the papers by Glashow and Coleman, among others.

Undoing science?

The idea that we should "undo" the Lorentz invariance, "undo" the energy-mass equivalence, or anything like that is simply an idea to return physics 100 years into the past. It is crackpotism - and a physics counterpart of creationism. The experiments that could have been interesting in 1905 are usually no longer so interesting in 2005 because many questions have been settled and many formerly "natural" and "plausible" modifications are no longer "natural" or "plausible". The previous sentence comparing 1905 and 2005 would be obvious to everyone if it were about computer science - but in the case of physics, it is not obvious to many people simply because physics is harder to understand for the general public.

But believe me, even physics has evolved since 1905, and we are solving different questions. The most interesting developments as of 2005 (for readers outside the Americas: 2006) are focusing on significantly different issues, and whoever describes low-energy experiments designed to find "10^{-7}" deviations from "E=mc^2" as one of the hottest questions in 2005 is either a liar or an ignorant. It is very fine if someone is doing technologically cute experiments; but their meaning and importance should not be misinterpreted.

#### snail feedback (15) :

Lubos:

Does your textbook ever teach you about the Sagnac Effect? I wish that you would be able to immediately say some opinion regarding the Sagnac Effect, without having to first look up in wikipedia what exactly is it. You can't talk about special relativity and Lorentz Invariance without addressing the Sagnac Effect. Unfortunately it is deliberately omitted from virtually all textbooks.

Quantoken

Happy New Year to everyone.

Quantoken, I agree that the Sagnac effect should be discussed as an example in courses of special relativity or even better general relativity, and properly calculated in both rotating and non-rotating reference frame. One reason is that it is useful in current technologies like GPS. Second reason is that some people ;-) tended (and tend?) to misinterpret certain principles and they thought (think?) that the experiment contradicts relativity which of course is not true.

For others, the Sagnac Effect is a shift of the interference pattern in the Michelson-Morley type of interferometer caused by rotation. Special relativity, of course, does not apply to rotating frames of reference without modifications, and the modifications - or equivalently, the gravitational field in GR associated with the rotation - shift the pattern proportionally to the angular velocity.

So, your point is that Lorentz invariance is exact and universal, similar to the rotational and translational invariances. I would like to demonstrate that this statement leads to a logical contradiction.

As you know, 9 inertial transformations mentioned above
together with time translations form the Poincare group. According to Wigner, there is a unitary representation of this group in the Hilbert space of any isolated physical system. The generators of this representation must satisfy the commutation relations of the Poincare Lie algebra. Since translations, rotations, and boosts are assumed interaction-independent, their generators are the same as in the non-interacting system. In the Poincare Lie algebra, the commutators of generators of boosts and translations are proportional to the generator of time translations, i.e., the Hamiltonian. Therefore, the Hamiltonian must be interaction-independent too. So, your assumption (Lorentz transformations are exact and interaction-independent) can apply only to non-interacting systems.

You're being completely silly. The boosts of course depend on the interactions (they are much like the Hamiltonian in this sense), and your argument is completely flawed. It would indeed be a disaster if we could not construct interacting Lorentz-invariant theories.

All theories we consider seriously - such as the Standard Model - are Lorentz-invariant.

OK, so you agree that boost generators depend on interaction. Then I want you to make a clear distinction between the notions
of "Lorentz invariance" and
"Lorentz transformations".

By Lorentz invariance (Poincare invariance is a better term, actually) I understand the idea that quantum descriptions of the system by different inertial observers are related by unitary transformations that form a representation of the Poincare group. This idea follows directly from the postulate of relativity, so I accept it 100%. You are right that all our theories are bound to be
Poincare invariant.

"Lorentz transformations" means a specific (linear) way in which certain observables transform when viewed from different reference frames. For example, there are Lorentz transformations for the
energy-momentum 4-vector of a particle. In quantum mechanics, these transformations should be obtained by applying the unitary boost transformation to the operators of momentum and energy of the particle. We have agreed, that boost generators must be interaction-dependent. Therefore, for interacting particles their energy-momentum transformations must depend on interaction. These transformations can take the simple linear Lorentz form only if the interaction can be neglected.

In summary, any interacting theory must be Poincare invariant, however boost transformations of observables must be different from simple linear Lorentz formulas.
Linear Lorentz transformations for time and position of events are just an approximation. So is the notion of the Minkowski spacetime.

No, Eugene, every step that you try to make seems to be flawed.

The boost generators are dynamical generators and consequently they depend on the interactions, but the transformation of the energy-momentum vector under the Lorentz group is unaffected by the existence of interactions.

It follows directly from the Poincare group. The Poincare group contains commutators like

[J_{mn},P_l] = delta_{lm} P_n - delta_{ln} P_m

and these commutators do imply that the energy-momentum vector transforms under the boosts in the same way as in the non-interacting theory.

These commutators hold in the free theory, but they also hold in the interacting theory where P_0 and J_{0i} receive interaction corrections.

Think about it for 10 minutes, it's not so terribly difficult.

I totally agree with you when you interpret P_l and
J_mn as operators of TOTAL observables (total momentum, total angular momentum, etc) of the isolated system. Total energy-momentum 4-vector does transform by linear Lorentz formulas.

However, the situation is different when you are trying to find the transformation rules for the energy and momentum of one particle in the interacting system of particles. In this case, the generator of boost is
the (total) operator that is a sum of one-particle boost operators (which I denote here by j_{0n}^i) plus interaction terms W_n

J_{0n} = j_{0n}^1 + j_{0n}^2 + ... + W_n

The operator that we are transforming is the one-particle momentum, e.g. p_l^1. So, we need to use the commutator

[J_{0n}, p_l^1]

which is different from

-delta_{ln} p_0^1

because p_l^1 does not (generally) commute with W_n.

Although, the total momentum of a compound system does transform by simple Lorentz formulas, the transformation laws for the momenta of individual particles are non-linear, they are interaction-dependent.

Dear Eugene,

it would be better if you were thinking a bit longer before you write something.

All vectors, including individual terms contributing to the energy-momentum vector, transform in the same "linear" way under the boosts because the commutator

[J_{mn},V_l] = delta_{lm} V_n - delta_{ln} V_m

actually holds for ALL vectors V_l.

Best
Lubos

Dear Lubos,

if we accept your statement for a moment, then we should conclude that the boost interaction W_n commutes with momenta of all individual particles. By the same argument, it also commutes with position operators of all particles. Then W_n must be simply a constant. Then by your own formula
the interaction in the Hamiltonian (= commutator of W_n with the total momentum P_n) must be zero. So, your statement is true only for non-interacting theories.

The contradiction can be avoided only if we accept that energy-momentum of individual particles in the interacting system are not 4-vectors and do not transform by linear Lorentz formulas.

Eugene, you are an infinite generator of flawed ideas.

The interaction part of the boost operator - corresponding to the interaction between two well-separated particles - indeed commutes with all individual momenta because the interaction between two well-separated particles is zero.

If it is not zero, then you cannot define the individual momenta of the separate particles because the particles are not separate. If you want to describe the strongly interacting particles as separate nevertheless, then you cannot neglect the fact that the interaction is caused by a field, and you should not forget to transform the field by the Lorentz transformations properly which you are not doing.

It is indeed inconsistent with relativity to imagine that two particles instantenously interact by a force that acts at a distance. Interactions in relativity are via fields that never propagate faster than light. Am I the first one who tells you this fact?

I thought that you have already agreed that the statement that the Lorentz symmetry and linear Lorentz transformation rules for momenta only hold in noninteracting theories is a stupidity - so why do you repeat the stupidity once again?

best
LM

OK, let's talk about relativistic QFT, if you like. It is not much different from what I said above. There is still Hilbert (Fock) space in which an unitary representation of the Poincare group is defined. Time translations are still generated by the full Hamiltonian whose interaction V is a certain combination of quantum fields. The interaction operator V does not commute with momenta and positions of individual particles. This is the reason why the time evolution of these observables is non-trivial. That's what we call interaction.

By the same argument, the interaction W_n in the total boost operator does not commute with momenta and positions of individual particles. Therefore boost transformations of these observables must be non-trivial, i.e., interaction-dependent.

What am I missing here? Are you saying that observables of individual particles (momenta, positions, spins) are no longer defined when the particles interact with each other? To me this looks like a very strong deviation from the postulates of quantum theory. These quantities can be easily measured, e.g., in the classical low-energy limit. So, there should be well-defined Hermitian operators corresponding to them. Moreover, if you deny the observability of
properties of interacting particles, then you can no longer talk about their Lorentz transformations. Then, again, you are not allowed to talk about Lorentz transformations for the time and position of events that involve interacting particles. Then, as I said, the concept of Minkowski spacetime (interval and all that) is an approximation which neglects particle interactions.

Dear Eugene,

don't you think that it would be a better idea to sit down and try to study quantum field theory, instead of filling my blog with utter nonsense?

There are no "operators of position of individual particles" in quantum field theory, and the rest of your posting unfortunately depends on this nonsensical assumption.

Best
LM

Just one more thing. You use the argument that special relativity forbids superluminal interactions in order to "disprove" my statement that interactions affect Lorentz transformations of space-time coordinates of events. Actually, in special relativity the assumptions of universal applicability and interaction-independence of Lorentz transformations is used to "prove" the impossibility of superluminal propagation of interactions. So, your logic seems to be backward, and I do not accept it.

Eugene.

Dear Lubos,

you say: "There are no "operators of position of individual particles" in quantum field theory"
Why not? There are Newton-Wigner position operators that can be defined for each massive particle in each subsector of the Fock space. I hope you wouldn't deny that particle positions are measured in numerous experiments with photographic plates, bubble chambers etc.
This suggests that there should be corresponding quantum operators. Doesn't it?

I agree that operators of position are not required when we are interested only in high energy scattering events, but all physics is not described by the S-matrix alone. If you deny the observability of position, how can you talk about space or spacetime?

Eugene.