Noether's theorem is a fundamental result in physics stating that every symmetry of the dynamics implies a conservation law. It is, however, deficient in several respects: for one, it is not applicable to dynamics wherein the system interacts with an environment; furthermore, even in the case where the system is isolated, if the quantum state is mixed then the Noether conservation laws do not capture all of the consequences of the symmetries[1].
In SR, force-free motion in an inertial frame of reference takes place along a straight-line path with constant velocity. Viewed from a non-inertial frame, on the other hand, this path of motion will be a geodesic curve in a flat spacetime. Einstein made the plausible assumption that this geodesic motion also holds in the non-flat case, i.e. in a spacetime region for which it is impossible to find a coordinate system that leads to the Minkowski metric in SR[2].
All spacetime models can be expressed in terms of the gμν = {4x4} matrix, differing only in the distribution of matrix elements. The gμν of Minkowski spacetime is the unit diagonal matrix {1 -1 -1 -1}; the gμν of Riemann spacetime is { X }. If a new spacetime model is introduced gμν={a0,-a1,-a2,-a3}, which is a non-unit diagonal matrix. (ds)^2=(a0)^2+(a1)^2+(a2)^2+(a3)^2, always holds, interpreting it as a non-uniformly flat spacetime, generalised Minkowski spacetime, and no longer a curved spacetime. Should Noether's theorem maintain its validity in this case.
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References
[1] Marvian, I., & Spekkens, R. W. (2014). Extending Noether's theorem by quantifying the asymmetry of quantum states. Nature Communications, 5(1), 3821. https://doi.org/10.1038/ncomms4821 ;
[2] Rowe, D. E. (2019). Emmy Noether on energy conservation in general relativity. arXiv preprint arXiv:1912.03269.
The short answer to the question is, Yes. However the statement regarding Noether's theorem and mixed quantum states is incorrect. There's nothing about mixed quantum states that stops Noether's theorem from applying.
For physical systems that aren't subject to fluctuations, Noether's theorem establishes the relation between symmetries-transformations that leave the equations of motion invariant-and conservation laws. In the presence of fluctuations the conservation laws are expressed as relations between correlation functions.
Flat spacetime is invariant under global Lorentz transformations and this leads to the conservation of the energy-momentum tensor. It is known how to describe quantum systems, in flat spacetime. Noether's theorem implies relations between correlation functions.
Curved spacetime is invariant under diffeomorphisms, an infinite-dimensional group of transformations. These are transformations that leave Einstein's equations invariant. What they imply for qantum matter in such geometries is partly known. What they imply for spacetimes that aren't solutions to Einstein's equations isn't known. The difficulty is that the group of diffeomorphisms is a non-compact group of gauge transformations and it is this property that makes it different from the non-compact Lorentz group of flat spacetime, that acts globally. It isn't known how to describe a quantum theory with a non-compact gauge group.
Dear Chian Fan
The image above is a schematic representation of a small part of spacetime. It shows the quantization of the volume of the universe: actually space itself. Time is the ongoing transformation of the variable properties of every unit of quantized space. Experiments have confirmed our observation that the volume of quantized space is invariant everywhere in the universe. That means that the transformations (actually energy) within the structure of quantized space are restricted to the surface area between all the units. The consequence is that every unit transforms its shape with the help of the transfer of a flux of infinite small volumes inside its boundary (smooth deformation). These transformations are synchronized with the transformation of the shapes of all the other units of quantized space (topological transformations under invariant volume). However, the image above is a schematic representation of absolute reality.
“Our” reality differs from absolute reality because we are only aware of the differences between the shapes of the units of the structure (the mutual relations). Einstein’s theory of special and general relativity isn’t about absolute reality, it is about “our” mutual relation grounded reality (phenomenological physics). Thus relative time is about the rate of change of compositions of energy under different conditions. Curved spacetime – general relativity – doesn’t even represent phenomenological reality because transformations under invariant volume cannot result in a curvature of quantized space.
The image above is not detailed so we have to incorporate quantum field theory (the basic quantum fields). Now the same image represent the general structure of the basic quantum fields: the universal electric field (3D topological field), the corresponding magnetic field (vector field) and the Higgs field (universal scalar field). The image only shows the electric field, because in absolute geometrical symmetry every scalar of the Higgs field is the inscribed sphere of every unit (“thus every cube has a scalar inside”). Moreover, the electric field and the Higgs field can exchange volume (Higgs mechanism) so the electric field represents the deformed parts of spatial units (quantized space) that have an internal spherical shape forming mechanism. That means that the difference between a scalar and the deformed part of the volume of 1 unit is determined by the distinct “configuration” of the internal spherical shape forming mechanism.
Every serious theory about the evolution of the universe (dynamic or static) describes a period that there wasn’t matter in the universe. Even Einstein admitted that without matter there is no theory of general relativity. The consequence is that gravity isn’t a basic quantum field, it is an emergent force field that comes to live at the moment matter is created in the early universe. Thus Einstein’s spacetime doesn’t describe absolute reality, the description is about phenomenological reality.
In other words, all the dynamics in the universe originates from the universal electric field and the corresponding magnetic field. Because without matter all the scalars of the Higgs field have exactly the same magnitude (flat Higgs field).
There a 2 universal conservation laws, energy and momentum. Energy is “generated” by every unit of quantized space and results in the changes of the shape of every unit. Momentum isn’t about a singular (elementary) property because momentum originates from classic dynamics and it means a certain local amount of energy and its influence (vector) on the other local amounts of energy around. But the conservation of the energy part of momentum is already covered by the law of conservation of energy. Every quantum of energy has momentum thus both universal conservation laws show that all the dynamics in the universe originate from the universal electric field and the corresponding magnetic field.
“Our” reality is dynamical reality (topological transformations under invariant volume). Thus symmetry like we know it is dynamical symmetry, created by the universal electric field and the corresponding magnetic field. All the changes are conserved (energy and momentum) thus all the geometrical and dynamical symmetries are conserved too.
With kind regards, Sydney
Dear Stefano Quattrini Stam Nicolis
Thank you very much for your replies. But still have some questions:
1) "Every symmetry of the dynamics implies a conservation law. not necessarily vice-versa ...." Can there be an example of this?
2) If forcing Noether's theorem to be absolutely fundamental and absolutely universal, what are the implications for general relativity? Will GR no longer hold? Or what would need to be modified?
Best Regards, Chian Fan
Dear Sydney Ernest Grimm
I don't quite understand your description. Is the universe made up of space-time plus three kinds of fields? Electric, magnetic and Higgs fields? What do they have to do with spacetime? Are you saying that energy-momentum is conserved everywhere in this model?
Best Regards, Chian Fan
Dear Chian Fan
Yes, that is correct! Spacetime is classic physics. The basic quantum fields tessellate the whole volume of the universe ("background fields"). So if you describe gravity in terms of quantum theory Einstein's theory of special and general relativity will be abandonned.
But of course special relativity wil be still used to synchronize events at different positions in outer space (and high energy particle physics, etc.). So actually nothing will change because at the moment we also use Newtonian gravity to calculate mutual influences between celestial bodies. General relativity is too complicated for "easy calculations".
If you want to create a theory of everything - inclusive quantum gravity - you have to "refresh" quantum field theory too. Because at the moment the whole description is too complicated (it is a patchwork of previous theories).
With kind regards, Sydney
Dear Chian Fan ,
so far the only reasonable attempt to improve General Relativity, by including field energy, is the following...
Article Modified general relativity and dark matter
negative energy was not an insurmountable problem it seems for Dirac. To my understanding it is..
Dear Chian Fan ,
something invented to guarantee the Lorentz Invariance...
Chian Fan
Every-continuous-symmetry implies a conservation law and vice versa. Noether's theorem isn't valid for discrete symmetries, because among the assumptions is that the transformations define a continuous group.
There's no ``forcing'' of Noether's theorem, regarding general relativity; indeed, it was ``discovered'' within general relativity!
I, already, recalled what happens for general relativity, where the symmetry group is a gauge group (that of diffeomorphisms): The conservation laws are defined in terms of covariant derivatives and conserved charges depend on the observer, since, in general, they can only be defined locally (and observers are local). If the spacetime is asymptotically flat, then conserved charges can be defined at infinity.