Do we need to find a motivation for symmetry: {?} → {invariance} → {conservation} → {symmetry} →
Should there be an ultimate symmetry that is identical to the conservation, structure invariance, and interaction invariance of the energy-momentum primitives and that determines all other symmetries?
Symmetry, invariance, and conservation are, in a sense, the same concept [1][2][3] and will generally be described in this order, as if symmetry were dominant.
As commonly understood, energy-momentum conservation was the first physics concept to be developed. It exists as a matter of course in mechanics, thermodynamics, and electricity. However, after physics entered the twentieth century, from quantum mechanics to general relativity, the conservation of energy-momentum has been repeatedly encountered with doubts [5][6][7][8][9][10], and so far it still can't be determined as a universal law by physics. Some of the new physics is insisting on "something out of nothing"[11][12][13][14] or spontaneous vacuum fluctuations[15], which equals to the rejection of energy-momentum conservation. The important reasons for this may be: First, Energy-momentum conservation cannot be proved† . Second, energy-momentum in physics has never been able to correspond to a specific thing, expressed by a unified mathematical formula‡, and it can only be the "equivalence" of various physical forms that are converted and transferred to each other [16]. Third, we have a biased understanding of the status of energy-momentum conservation, such as "These symmetries implied conservation laws. Although these conservation laws, especially those of momentum and energy, were regarded to be the most important of all. Although these conservation laws, especially those of momentum and energy, were regarded to be of fundamental importance, these were regarded as consequences of the dynamical laws of nature rather than as consequences of the symmetries that underlay these laws."[17]. Conservation of energy-momentum was relegated to a subordinate position. Fourth, it is believed that the Uncertainty Principle can be manifested as a " dynamics ", which can cause various field quantum fluctuations in the microscopic domain, and does not have to strictly obey the energy-momentum conservation.
"Symmetry" refers to the "invariance under a specified group of transformations" of the analyzed object [4]. Symmetry is always accompanied by some kind of conservation, but conservation does not only refer to the conservation of energy-momentum, but also, under different conditions, to the conservation of other physical quantities, such as charge, spin, or the conservation of other quantum numbers. Thus, "conservation" is usually the constant invariance of something at some level, and Wigner divided symmetries into classical geometrical symmetries and dynamical symmetries, which are associated with specific types of interactions, every interaction has a dynamical invariance group. "It may be useful to discuss first the relation of phenomena, laws of nature, and invariance principles to each other. This relation is not quite the same for the classical invariance principles, which will be called geometrical, and the new ones, which will be called dynamical."[1]. According to Wigner, we can define the "geometric invariance" of everything as the manifestation of interactions filtered through the absoluteness of the spatio-temporal background. This interaction exhibits itself whenever you assume an observer*. displacement invariance, Lorentz invariance are typical. We can define all "dynamical invariance" as manifestation when the background absolutes of the potential field are filtered out. gauge invariance, the diffeomorphism invariance are typical manifestations." from a passive role in which symmetry is the property of interactions, to an active role in which symmetry serves to determine the interactions themselves --a role that I have called symmetry dictates interaction." "Einstein's general relativity was the first example where symmetry was used" actively to determine gravitational interaction" [2]. This expresses the same idea, that the role of symmetry is elevated to the status of "force". Gross says that the secret of nature is symmetry. The most advanced form of symmetries we have understood are local symmetries-general coordinate invariance and gauge symmetry. The most advanced form of symmetries we have understood are local symmetries-general coordinate invariance and gauge symmetry. unified theory that contains both as a consequence of a greater and deeper symmetry of which these are the low energy remnants [18]. He regards the unification of general relativity and quantum field theory as a unification of symmetries. He regards the unification of general relativity and quantum field theory as a unification of symmetries. If we define generalized invariance as the completeness of the structure, properties, and laws of interaction of the analyzed objects when they interact, i.e., the undecomposability of the whole as a whole, the conservation of the properties (charge, spin, other quantum numbers, etc.), and the consistency of the interaction relations (laws), it is clear that the invariance in this case is special invariance, which means only the invariance of the laws of interaction.
While symmetry, conservation, and invariance are almost equivalent expressions at the same level, there are subtle but important differences. If unbounded, it is the order in which the three are expressed, who actually determines whom, and who ultimately determines the laws of physics. In any case, when we currently speak of symmetry, it must correspond to specific invariance and conservation, not to broad invariance and conservation. This in fact greatly limits the claim that "symmetry dictates interaction", since interaction is much more general. There is no such thing as a failure of interaction, but there is often a failure of symmetry, unless we decide that there will be an ultimate symmetry that determines all other symmetries.
"A symmetry can be exact, approximate, or broken. Exact means unconditionally valid; approximate means valid under certain conditions; broken can mean different things, depending on the object considered and its context. different things, depending on the object considered and its context."[19] "It is not clear how rigorous conservation laws could follow from approximate symmetries"[1]. This reflects the uncertainty of the relationship between conservation currents ( charges) and symmetries, and if we know that conservation currents can still be maintained even with approximate symmetries, it should be understood that this must be a function of the fact that conservation currents have a more universal character. From a reductionist point of view, the conservation charge at all levels will gradually decompose with the decomposition of matter, until finally it becomes something that cannot be decomposed. Such a thing can only be the most universal energy-momentum and at the same time be the ultimate expression that maintains its conservation as well as the invariance of interactions. Otherwise, we will pursue the questions:
1) If energy-momentum conservation is not first, where does the power to move from one symmetry to another, symmetry breaking [11] [12], come from? How can symmetry violations [13] in physics be explained?
2) If symmetry fully expresses interactions, how do we evaluate "symmetry implies asymmetry", "imperfect symmetry", " approximate symmetry", " hidden symmetry"? hidden symmetry"?
3) One of the implications of energy-momentum conservation is that they have no origin, are a natural existence, and do not change with scale and energy level or temperature; symmetry has an origin, and is related to scale, temperature and energy level. How are they equivalent to each other?
4) Must there be an ultimate symmetry which will determine everything and be consistent with conservation and invariance?
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Notes
† We will analyze this separately, which is its most important physical feature [20].
‡ Can different forms of energy be unified?[16]
* We can define the actual observer to be the object of action and the abstract observer to be the object of action for analysis. For example, when we analyze the Doppler effect, we are analyzing it in the abstract; if you don't actually detect it, no Doppler effect occurs in the object of analysis.
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References
[1] Wigner, E. P. (1964). "Symmetry and conservation laws." Proceedings of the National Academy of Sciences 51(5): 956-965.
[2] Yang, C. N. (1996). "Symmetry and physics." Proceedings of the American Philosophical Society 140(3): 267-288.
[3] Yang, C. N. (1980). "Einstein's impact on theoretical physics." Physics Today 33(6): 42-49.
[4] Brading, K., E. Castellani and N. Teh (2003). "Symmetry and symmetry breaking."
[5] Bohr, N., H. A. Kramers and J. C. Slater (1924). "LXXVI. The quantum theory of radiation." The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 47(281): 785-802.
[6] Dirac, P. A. M. (1927). "The quantum theory of dispersion." Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 114(769): 710-728.
[7] Carroll, S. M. and J. Lodman (2021). "Energy non-conservation in quantum mechanics." Foundations of Physics 51(4): 83.
[8] Bondi, H. (1990). "Conservation and non-conservation in general relativity." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 427(1873): 249-258.
[9] Maudlin, T., E. Okon and D. Sudarsky (2020). "On the status of conservation laws in physics: Implications for semiclassical gravity." Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 69: 67-81.
[10] Pitts, J. B. (2022). "General Relativity, Mental Causation, and Energy Conservation." Erkenntnis 87.
[11] Hoyle, F. (1948). "A new model for the expanding universe." Monthly Notices of the Royal Astronomical Society, Vol. 108, p. 372 108: 372.
[12] Vilenkin, A. (1982). "Creation of universes from nothing." Physics Letters B 117(1): 25-28.
[13] Josset, T., A. Perez and D. Sudarsky (2017). "Dark energy from violation of energy conservation." Physical review letters 118(2): 021102.
[14] Singh Kohli, I. (2014). "Comments On: A Universe From Nothing." arXiv e-prints: arXiv: 1405.6091.
[15] Tryon, E. P. (1973). "Is the Universe a Vacuum Fluctuation?" Nature 246(5433): 396-397.
[16] Chian Fan, https://www.researchgate.net/post/NO10_Can_different_forms_of_energy_be_unified
[17] Gross, D. J. (1996). "The role of symmetry in fundamental physics." Proceedings of the National Academy of Sciences 93(25): 14256-14259.
[18] Gross, D. J. (1992). "Gauge theory-past, present, and future?" Chinese Journal of Physics 30(7): 955-972.
[19] Castellani, E. (2003). "On the meaning of symmetry breaking." Symmetries in physics: Philosophical reflections: 321-334.
[20] https://www.researchgate.net/publication/369890893_Summary_Table_of_Light_String_and_Light_Ring_Properties_Supersymmetry
Dear Professors
Let us assume that James Webb Space Telescope, or JWST, data allows the general relativity zero-active-mass condition, or the condition that particles cannot gain energy-momentum from the expansion of spacetime in an expanding universe, is true. This enables the conservation of energy-momentum to have a cause that is, rest-mass energy is gravitational binding energy [1].
“Second, energy-momentum in physics has never been able to correspond to a specific thing, expressed by a unified mathematical formula.”
Now we do have a “unified mathematical formula” defined in [1], for energy-momentum conservation.
This new gravitational binding energy concept separates inertia from momentum conservation allowing inertia to be a local vacuum Lagrangian interaction that is useful [2].
[1] https://www.degruyter.com/document/doi/10.1515/zna-2023-0168/html
[2] https://www.researchgate.net/publication/374170383_Thrust_Capacitor_Design_and_Theory
Thank you
It is true that there is a give-and-take between observation and theory; observations suggest conservation laws and knowing that conservation laws reflect the existence of symmetries, imply the relevance of symmetries that, in turn lead to new predictions. It's the property that we don't know everything about natural phenomena, that our description is the result of successive approximations that manifests itself as the realization that symmetries just describe our state of knowledge at any given time. As we learn more and in finer detail about Nature, it can turn out that certain symmetries may turn out to be approximations and that new symmetries may turn out to be relevant. So then the question arises how are these symmetries related.
And, indeed, it is useful to distinguish history from logical content. How notions were discovered isn't the most useful way for understanding their meaning.
Regarding the questions:
1) Conservation laws imply symmetries and vice versa. Conservation of energy and momentum is the expression of invariance under translations in time and space. Now one can ask, how might it be possible to break these symmetries.
The way to break invariance under translations in space seems quite easy: That's the meaning of Newton's 2nd law, that the variation in time of momentum is equal to the force that another system, external to the system of interest, exerts on it.
So it's necessary to be able to discuss two systems and have a notion of how they can interact-how one can exert a force on the other.
Breaking invariance under time translations means that the system can know ``how old it is'', it doesn't know, only the difference between two moments in time, but it knows about each moment individually. In Newtonian mechanics this property can be shown to be described by so-called ``non-conservative'' forces, which are those forces, that can't be expressed as the gradient of a scalar function of the position; forces that can be expressed as the gradient of a scalar function of the position are known as ``conservative'' forces.
To go from there to discussing general relativity requires some more effort, however. One, first, has to understand what happens in special relativity, that generalizes Newtonian mechanics. That generalization isn't as drastic as one might think, because, in special relativity, too, invariance under time translations and translations in space are global symmetries; they're just imposed on objects moving in spacetime, not space and time separately and there is the additional requirement of invariance under Lorentz transformations. But everything else is as before and, if there is more than on system, then they can interact, only, now, the requirement is that the interaction be invariant under Lorentz transformations.
The simplest example is that of a relativistic, charged, particle, in an electromagnetic field.
It is here that a new kind of conservation law appears-that of the conservation of electric charge. And that-global-Lorentz invariance implies that charge can't be conserved by instantaneously ``disappearing'' from one place and ``appearing'' some place else: If the charge density varies in time, this means that there's a current, that flows from one place to the other and the charge density and the current are components of a ``larger'' structure the current 4-vector that must be conserved, for Maxwell's equations to be consistent. What is interesting is that it's possible to show that requiring global Lorentz invariance, 4-vector current conservation and that the equations don't contain more than two derivatives, leads to Maxwell's equations as the only equations that satisfy these requirements.
Chian Fan ,
it is the fundamental tool that was first summarized by Helmolz and after him many others. It is currently used in every experiment with particles and in any branch of mechanics and thermodynamics.
that simply suggests that what is called a vacuum cannot be the emptiness as conceived in relativity...
Space-time and Lorentz invariance come from Einstein's interpretation of the Lorentz Transformations where he assumed the equivalence of inertial frames. The isotropy of the speed of light hence the universal constancy of c for any observer is a consequence.
Einstein had to assume by "stipulation" that the speed of light is constant for any observer, so light has a very special behavior that singles out any preferred frame or propagation frame, which allows the equivalence of inertial frames.
That is well known to be at variance with what was strongly supported by experts in electromagnetism like Maxwell, Hertz, Lorentz, Larmor, Heaviside etc.
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Let's suppose the validity of the equivalence of inertial frames within the same physical problem.
It is well known experimentally that the RADAR Doppler increases its frequency when bodies approach. From the inertial frame of the RADAR station, the detected frequency increases, that is how that system works as experimentally verified on a day-to-day basis.
If the relative speed with the reflector is maintained, considering it inertial, by applying the Einstien-Lorentz Transformations, the result is that
the RADAR station measures progressively higher energy available due to higher energetic quanta of radiation detected.
If the reflector does not decelerate, at every bounce of the EM waves, some energy is created out of nothing.. that is in clear infringement of the conservation laws, it is a perpetuum mobile of the first kind.
In order to get rid of that one must assume that at least the reflector can decrease its speed and hence it is non-inertial.
One of the two objects must decrease its speed in order to transfer its kinetic energy to the radiation. So the effect cannot occur between inertial frames.
Basically light and two inertial frames in the same physical problem are something not possible since energy conservation laws forbid. So what can be obtained from LT, in Einstein's interpretation, is just a first order approximation of the effect...
The Lorentzian interpretation of the Lorentz Transformation, do not involve symmetries, hence the reason of the violation of the conservation laws is removed in such case.
Continuing with @Chian Fan questions:
2) Once a symmetry has been identified, it is possible, on the one hand, to erite down the interactions that are consistent with it, as well as the interactions that are not-that's where the mathematical properties of what's a group become relevant. And then it's possible to describe the effects of each and, therefore, to deduce bounds on the coefficients of the different contributions.
3) Once more, the origin of the conservation of energy and momentum is the invariance of the description of natural phenomena under translations in time and translations in space.
The issues of scale deserve a separate answer.
All questions like “what comes first” - and first of all the so-called “fundamental question of philosophy” (forgive me, Lord!) - do not give rise to anything other than many years of empty word debate. And then - conflicts of varying intensity, up to revolutionary wars. There is nothing new under the Moon: let us remember Hegel, for whom Being through Nothing (through singularity) gives rise to Otherness. And vice versa. And so it is with all other questions of this kind. We need to look for connections, and not the mythical “primacy”, which does not exist according to definition.
It is pointless to speak about these quite general topics if the fundamental point here is that the necessary coherence with known experiments is missing.
If our point of view is the classical point of view (phenomenological physics) it is impossible to state that there exists true conservation of properties.
Only if we use the concept that the volume of the universe is tessellated by a number of basic quantum fields and nothing else we can state that the mechanism behind the conservation of properties exists everywhere in the universe. The same for the existencce of invariance at the smallest scale size.
With kind regards, Sydney
The actual reason of the principle of energy conservation is to exclude the possibility (till contrary proof) of transferring a non-finite amount of work (energy) from a certain object in space to another (meaning extract a net amount of energy out of nothing), avoiding a perpetuum mobile of the first kind.
A prerequisite for addressing these issues is a mastery of the undergraduate curriculum of physics-either at a university, or by oneself. There's no point pretending to live in the 20th century, already-we're living in the 21st century. A good introduction can be found here: https://www.feynmanlectures.caltech.edu/
There's no point trying to reconstruct what has been understood in the last 400 years from scratch, nor pretend this hasn't happened.
So going on with the questions of Chian Fan
What wasn't addressed in my answer to question 3) was the issues of scale and temperature (``energy level'' doesn't make sense in this context).
The statement that a physical system is in equilibrium at temperature T means that the quantities of interest have a certain probability distribution, namely exp(-E/kBT)/Z, where E is the energy of the system in the configuration of interest and it's assumed that how the energy depends on the quantities of interest. Then it's possible, in principle, to compute the moments of the quantities of interest and, in many cases, also, in practice, using numerical or analytical techniques.
So the conservation of energy and momentum in this case means that, indeed, the system, in equilibrium at temperature T, is invariant under translations in time and space; only that the expression of this invariance is realized in terms of relations between correlation functions, defined through a probability distribution.
The statement about how the conservation of energy and momentum depends on the scale under study is similar: Energy and momentum are conserved, at any scale one studies; at scales where quantum effects become relevant, what happens is, in fact, similar to what happens when the system is in equilibrium at a finite temperature-only the way the probabilities are calculated changes, as does the space of states.
It would be nice that you understand what happened instead of purporting that it has been understood...
What I report is an infringement of conservation laws for two inertial systems exchanging light...and it is not controvertible...it is enough to perform some not-so-complicated calculations to understand it...
Regarding question 4) the short answer is No, in the sense that ``ultimate symmetry'' doesn't make sense.
However, any symmetry, by definition, describes situations that don't change upon realizing certain transformations and this is expressed by conservation laws. In the presence of fluctuations-thermal, quantum, disorder-these laws are expressed in terms of moments of probability distributions.
What are the ``ultimate'' symmetries of nature isn't known. We know what the classical symmetry of spacetime is-invariance under diffeomorphisms-and we know that it can't be the symmetry of spacetime, when quantum effects become relevant. How it emerges, as the classical limit of the quantum description, isn't known. We know what the consequences of the invariance of spacetime under diffeomorphisms implies for describing observations in cosmology.
We know some of the symmetries of quantum matter, when gravitational effects can be neglected, but not everything, yet.
There's still a lot to learn, but that implies having understood what's, already, known: https://www.youtube.com/watch?v=IFBtlZfwEwM
Chian Fan,
All laws in physics with no exception are described by mathematical expressions.
Your question about symmetry and invariance and conservation laws is about how math teaches physics as elucidated by Distinguished Mathematical Physicist Peter Woit in this short Big-Think YouTube video https://youtu.be/c9tNoH6RB-k?si=SBAU0nVVqUii_oDm
Subtleties do enter the picture in quantum field theory-the quantum theory of infinitely many degrees of freedom-especially regarding scale invariance. In that case, however, the symmetries that are relevant are internal symmetries, not spacetime symmetries-since the only quantum field theories that are under calculational control describe quantum matter in flat spacetime and, while the energy-momentum is, always, conserved, it does play a role in describing the invariance-or its breaking-under scale transformations: Namely, if the energy-momentum tensor is traceless, then it can be shown that the theory is invariant under scale transformations. Quantum effects can lead to the breaking of scale invariance, which means that the corresponding conservation law receives corrections. The most famous example is QCD, that, classically, is invariant under scale transformations, but isn't, when quantum effects are taken into account. The way scale invariance is broken in this case is what allows understanding the behavior of hadrons at high energies and, in particular, the properties of quarks.
Similar subtleties ar encountered when coupling field theories to baths-whether quantum, thermal or disorder.
Momentum is a concept that originates from classic physics. It describes the influence of a phenomenon on other phenomena based on its energy content and the direction of its motion. The consequence is that every quantum of energy has momentum too.
Thus every quantum of energy is enveloped by the law of conservation of energy and the law of conservation of momentum. So if I subtract the “energy part” from the law of conservation of momentum – because it is already covered by the law of conservation of energy – I get a “new” conservation law: the law of conservation of vectors. Because the direction of every quantum of energy is determined by vectors. Vectors that correspond with the amount of energy of the quantum of energy. In line with the existence of the universal electric field and its corresponding magnetic field (together electromagnetic field) in QFT.
The nature of the quantum of energy is described in Einstein’s famous formula E = m c2. It means that we must apply surface area (c2) to every locked quantum of energy in the mass (m) to become a free quantum again in vacuum space. If energy is just volume Einstein’s formula should be E = m c3. However, we know that energy isn’t volume because if we supply energy to a particle – actually accelerating the particle – its volume doesn’t expand. A huge expansion of the size of particles that nears the speed of light – because of the huge energy supply – is never observed in particle colliders.
Now how must we interpret the quantum of energy as a fixed amount of surface area? Because there exists no surface area without a volume.
The idea that our universe has a spatial structure – a metric – originates from the ancient Greek philosophers. But it was also researched by Heisenberg, Flint, March, Goudsmit, Möglich and others during the first half of the 20th century. Because at the smallest scale size particles and the wave length of ultra short electromagnetic waves seem to share an underlying metric (≈ 1 x 10-15 m). The proposed metric is termed “minimal length scale” and can be easily proved with the help of the Planck-Einstein relation.
The existence of a spatial structure that manifests itself in the properties of the basic quantum fields (QFT) – determined with the help of experimental physics – is a given. Therefore, the consequence for the universal electric field is that it can only be described mathematically in a correct way as a topological field under invariant volume. That means that the variance of the universal electric field represents the mutual topological deformation of surface area between all the units of the spatial structure.
The topological deformation by units with equal basic properties results in all kinds of symmetries everywhere within the structure. Symmetries like Stam Nicolis described in his previous comment.
Anyway, the Standard cosmological model suggests that the volume of space isn’t invariant, although cosmologists have the awkward opinion that space within the boundary of a galaxy doesn’t expand “in the same rate” as space in between galaxies. Recent observations have confirmed that the space within clusters of galaxies don’t show the same rate of expansion too. The consequence is that the proposed “expansion of space itself” is just the drift away of “single” galaxies that results in the increase of the volume of the voids. In other words, the non-Doppler red shift of the light of distant galaxies has no direct relation with the proposed expansion of space itself.
This is really disappointing because some opinions about physical reality at smaller scale sizes are influenced by the conceptual framework of the Standard cosmological model. A model that seems to be a total mess if we include the new observations of the James Webb Telescope too.
With kind regards, Sydney
1) What kind of symmetry is unbreakable?
Dear Stam Nicolis
Thank you very much for your comment, I think I and others will benefit from it.
"Conservation laws imply symmetries and vice versa. Conservation of energy and momentum is the expression of invariance under translations in time and space. now one can ask, how might it be possible to break these symmetries." The question of the possibility of breaking energy-momentum symmetries, I haven't understood your answer yet, but your question must be a great question!
My point, however, is that we must have a symmetry that ultimately cannot be broken, and they can only be energy-momentum conserving symmetries. One might think "vacuum energy", but I don't know how physics describes its symmetry. I've always wondered about the concept of vacuum energy. If there is no way for energy-momentum to return to vacuum energy, then it should not exist. Should there also be symmetry between vacuum energy and non-vacuum energy?
Best Regards, Chian Fan
Dear Chian Fan
Symmetries are dynamical (like everything in the universe). So there are no "unbreakable" symmetries in the universe, related to local compositions of energy. There are only transformations so energy is constantly redistributed in the universe (like E = m c2 shows).
With kind regards, Sydney
Dear Sydney Ernest Grimm
Thank you very much for bringing your perspective and insights on the expansion of the universe. You previously recommended a site for updated physics information to me.
You said, "Momentum is a concept that originates from classic physics". Yes, but momentum took on a new concept when the photon appeared, and the relationship between E and P was determined from that.
I don't understand your "the law of conservation of vectors", because "vector" is not a quantity with a specific dimension. I also don't understand the meaning of "minimal length scale". I may need to dig deeper into your paper.
Regarding symmetry, you said "Symmetries are dynamical". Yes, but this is not arbitrary. I agree with Stam Nicolis that "Conservation laws imply symmetries and vice versa". As long as there is conservation of energy-momentum, there will be corresponding symmetries, unless conservation of energy-momentum is negated.
Best Regards, Chian Fan
Dear Chian Fan
Stam Nicolis is right, conservation laws can be seen as symmetries. The law of conservation of electric charge is a much smaller symmetry because it is about distinct properties of charged particles. In the early universe there was less matter than now so it is understandable that symmetries can increase and maybe decrease during the evolution of the universe. But all these “smaller” symmetries are manifestations of the universal conservation laws (energy and momentum). There are also geometrical symmetries and force symmetries, etc. (Every equation is a symmetry.)
A true vector is 1-dimensional and its magnitude is generated by the dynamics of the electric field. If you think: “This is totally confusing”, I agree. The problem is that the underlying “tangible” reality of QM isn’t known (https://www.nytimes.com/2019/09/07/opinion/sunday/quantum-physics.html).
Anyway, some months ago there was an article in Nature about the measured velocity of tunnelling. Tunnelling is the pass on of a particle through a barrier (under certain conditions). The velocity of the transfer of the particle from one side of the barrier to the other side exceeds the speed of light. It shows that the concentrated energy of the particle – that is restricted to the speed of light – didn’t pass the barrier (or at least only partly). So it was the vector configuration of the particle that passed on. But because the electric field and the magnetic field (vector field) are corresponding fields every quantum of energy generates a vector and visa versa. That is why the particle “pops up” again at the other side of the barrier. So vectors are “tangible” too, although vectors can only be observed with the help of matter (like iron powder can show the shape of the vector field of a magnet). The forced “split” between the electric field and the corresponding magnetic field is also visible in the famous experiment with the 2 slits.
The minimal length scale is a known concept in theoretical physics although its size is disputed. Most theorists have embraced the Planck units because it gives the necessary “space” for the Standard model of particle physics. If the minimal length scale is about the smallest detected particles and wave length the Standard model must be wrong. Now the asymptotic freedom must originate from the existence of the barrier of the “tangible” metric and it is easy to see that this is not an attractive idea for everyone. Unfortunately nobody has explained the lack of complexity between both proposed metrics.
I agree that the meaning of momentum has changed in modern physics. But only the proposed scale size because of the discovery of Planck’s constant (as you wrote above).
With kind regards, Sydney
Dear @Chian Fan
A short answer to your question is that ``global" symmetries-whose parameters don’t depend on the point in spacetime-can be broken, either explicitly, or ``spontaneously"-in he latter case, the equations of motion are invariant under the transformations, but their solutions are not; the typical example is the magnet, that displays spontaneous magnetization, i.e. behaves like a ferromagnet, that breaks rotation symmetry, below the Curie temperature and behaves like a paramagnet or diamagnet, that preserve rotation symmetry, above the Curie temperature. The rotation symmetry doesn't refer to rotating the material in space, but rotating the magnetic moments of the material. (In this sense it's an `ìnternal'' symmetry, not a ``spacetime'' symmetry.)
``Local" (also known as ``gauge") symmetries can’t be broken. In general relativity there aren’t any global symmetries, unless at infinity, assuming the spacetime is asymptotically flat. All symmetries in gravity are local, since (classical) gravity is defined by the property of invariance under general coordinate transformations.
@Sydney Ernest Grimm Conservation laws are the consequence of invariance of the equations of motion under continuous symmetries (in the absence of fluctuations); and of the partition function that describes the coupling with the bath (in the presence of fluctuations). In the latter case the symmetries and the conservation laws are expressed as identities between the moments.
And there is an important distinction between spacetime symmetries, such as translations, rotations, boosts or dilatations, that act on the coordinates of spacetime and internal symmetries that don’t act on the coordinates.
The conservation of charge isn't a ``smaller'' symmetry (there isn't any notion of ``larger'' or ``smaller'' symmetry)-it's an ``internal'' symmetry: It isn't the result of the equations being invariant under coordinate transformations, but under transformations of the fields. The transformations in question do depend on the point in spacetime, but don't affect the point in spacetime.
So the conservation of energy and momentum-in gravity-is, indeed, an example of a symmetry that can't be broken, because its breaking would imply inconsistency rather than incompleteness of the equations. The conservation of energy and momentum in gravity is the expression of a local symmetry and, indeed, this was discovered indirectly: Einstein and Hilbert, independently, noticed that the equations for general relativity seemed to include four equations more than they had thought necessary. So there was a debate whether the additional equations were independent equations or not and what they could mean. That was the question that Hilbert addressed to Emmy Noether-and she found that the four additional equations expressed the local conservation of energy and momentum and she found that this was true in an appropriate way for any symmetry and the corresponding conservation law for any physical system, by using the action principle for its description. Even though she obtained this result in 1918, it took a long time for it to become part of the physics curriculum.
(It's of historical interest that the properties that the four ``additional'' equations must be identities had been found-but in the context of Riemannian geometry-by a doctoral student of Felix Klein, called Bianchi-so these identities, in gravity, are, now, called ``Bianchi identities''; and it is remarkable that Hilbert, who was in Goettingen, didn't make the connection!)
Vacuum energy is another story. The reflexes that are now standard weren't then. So when Einstein found his equations and Hilbert wrote the action principle, it wasn't clear whether the equations were the most general equations or the action the most general action and what were the symmetries-Noether's work was, still in the future (1918) (or not so well understood that people would take it into account immediately, as happens now, when teaching physics). So Einstein found that he could introduce additional terms in his equations-describing what he called the cosmological constant-in order to support the claim of a static universe. There he was mistaken, as Willem de Sitter showed by actually solving the equations with such a term included: The solution isn't static, it describes a universe in accelerating or decelerating expansion, depending on the sign of the constant.
That it described the ``vacuum energy density'' of the spacetime was understood much later. Also that the reason Einstein was able to introduce it at all, without making the equations inconsistent implied that the cosmological constant is, in fact, inevitable, since it's consistent with the symmetries of general relativity. For a long time it wasn't possible to measure it and a lot of effort went into trying to show that its value must be zero. Only in 1998 was it possible to finally finish measuring it for our universe-and it was found what its value is and that its value is positive.
The question about ``dark energy'' is, whether it can be completely described only by the cosmological constant, or whether taking into account its effects requires introducing additional fields. For the moment, it seems that the cosmological constant suffices.
Dear Sydney Ernest Grimm
Thank you for the many ideas you brought，Such as "Every equation is a symmetry", "The forced 'split' between the electric field and the corresponding magnetic field", "the asymptotic freedom must originate from the existence of the barrier of the 'tangible' metric" . magnetic field", "the asymptotic freedom must originate from the existence of the barrier of the 'tangible' metric"...... , are all thought-provoking ideas.
But if all symmetries are breakable, is there a final state according to reductionism? What should the final state be? Should it be "invariant"? If it does not have invariance, then it can continue to be divided. This will lead to the "infinite division" dilemma.
Best Regards, Chian Fan
Dear Stam Nicolis
Thank you very much for providing the associated knowledge about symmetries: continuous symmetries, internal symmetry, spacetime symmetries, global symmetries and Local symmetries; Bianchi identities, Einstein's field equations, Hilbert's action principle, Noether's Theorem; vacuum energy, dark energy, etc... Some of the points you gave I am still learning and thinking about, such as, "That was the question that Hilbert addressed to Emmy Noether-and she found that the four additional equations expressed the local conservation of energy and momentum and she found that this was true in an appropriate way for any symmetry and the corresponding conservation law for any physical system, by using the action principle for its description. " Are you talking about Noether's Second Theorem here? (I haven't understood her original text).
* The structure and behaviour of the universe is hierarchical, and this hierarchy results from causality. Thus, in the same hierarchy, "symmetry = invariance = conservation = causality". We do not believe that nature is a mixture of "causality" and "randomness" because we cannot explain how these two conflicting modes of behaviour can be related to form the universe that is currently functioning according to a strict order. There must be an ultimate order dependence of the universe, which can only be energy-momentum conservation.
*"The cosmological constant is, in fact, inevitable, since it's consistent with the symmetries of general relativity." If the field equation is the final equation, I don't think there will be, nor should there be, any constant term, since it involves the question of who decides what value to take for this constant. Vacuum energy or not, dark energy or not, are they themselves causing spacetime expansion? Or is it the "difference" between the local gμν and the infinitely distant gμν? We have no idea what these two energies are, and whether there is an interaction between them and the known energy-momentum. If there are regular exchanges, then they should be "averaged" out, like a drop of ink in a basin of water. If they are not able to exchange with regular energy-momentum, they should not be called "energy".
You and Sydney Ernest Grimm both talk about equations. You say, "the equations of motion are invariant under the transformations, but their solutions are not", and Sydney Ernest Grimm says "Every equation is a symmetry". I need to rethink this.
Best Regards, Chian Fan
Dear Chian Fan
Your latest remark about reductionism was answered by some ancient Greek philosophers. They reasoned that there must be a minimal metric in the universe because without a minimal metric reality must be crazy complex in a "tangible way" everywhere in the universe. The concept is termed "minimal length scale" in physics and it is important in the search for quantum gravity and in research to "build up" physical reality from scratch (mathematical physics).
If is obvious that the minimal length scale corresponds with the metric of the structure of the universal electric field and corresponding magnetic field because both fields represent the 2 universal conservation laws (energy and momentum). Unfortunately there is little progress because the focus is on QM. Theorists like 't Hooft, Sean Carroll, Lee Smolin, Susskind, etc. have published papers about understanding "the tangible nature" of QM but there is no accepted break through.
With kind regards, Sydney