The approach of first principles has been pursued in the development and history of physics. Ever since the establishment of the Standard Model of particle physics in 1970s, the idea of going after theory of everything has become popular as the latest approach of first principles among theoretical physicists for unifying all particles and interactions. However, we seem to live in a dynamic world as indicated, e.g., since the discovery of an expanding Universe and it is definitely at odds with the static picture of an ultimate unified theory for physics.
The dynamic picture tells us that the time reversal symmetry has to be broken and it has to be the first (broken) symmetry. Whatever first principles we propose have to be able to naturally break this symmetry first in the very beginning. And there is no reason why the current 4-dimensional spacetime, in particular, its dimensions can’t be dynamic. It is probably more natural to consider that spacetime has evolved in a dimension-by-dimension way.
First of all, we propose and summarize the three first principles as follows:
- A measurable finite physical world is assumed.
- The quantum version of the variation principle in terms of Feynman’s path integral formalism is applied.
- Spacetime emerges via dimensional phase transitions (i.e., first time dimension and then space dimensions got inflated).
Measurable finiteness of the physical world is probably responsible for the appearance of the Universe. In other words, observation itself defines what we can actually observe. For a meaningful observation, at least due to human’s limitations, all physical quantities have to be finite. By “measurable” here we assume that one can do consistent measurements. For consistency of measurements, we have to introduce a well-defined symmetry group – the holonomy group that preserves the metric of the spacetime geometry. That is, symmetries emerge from the requirement of consistent observation. Later on, we’ll see how this principle coupled with the other two presents us a variety of emerging symmetry groups and particle fields in a dynamic world.
Many variants of the variational principle have been developed in the history of physics and math. For example, Fermat’s principle of least time for optics, and Hamilton’s principle or the principle of least action for classical mechanics. The quantum version of the variational principle was first proposed by Feynman using his well-known path integral formulation. This principle shows how to construct physical laws in mathematical form without invoking any ad hoc laws a priori. In other words, the variational principle essentially presents why math works in physics.
The first two principles might be good enough to construct a static or single-phase theory for a given physical system. But it does not introduce the physical contents nor the dynamics involving phase transitions. The third principle on spacetime evolution is critical for invoking varied particles and interactions at different phases of the dynamic Universe. First the time dimension was born. Then one space dimension got exponentially extended to make the Universe a 2-d spacetime. Finally two more space dimensions were inflated into our current 4-d spacetime. As a matter of fact, once the time dimension emerges first, the other two inflation processes will follow naturally under no further assumption.
Based on these three first principles, the new mirror matter theory can be described by a series of Supersymmetric Mirror Models (SMM1 for 1-d time, SMM2 for 2-d spacetime, and SMM4 for 4-d spacetime) and gravity can be understood as an emergent classical phenomenon of smooth spacetime geometry.
In the quantum variational principle, the quantum probability amplitude is defined as a coherent sum of contributions from all possible configurations. Here the contribution of each configuration is determined by an exponential factor of exp(iS/ħ) where ħ is the Planck constant and S is the action that defines all physics depending on involved symmetries and particle fields.
Feynman’s path integral formalism, in my opinion, is his greatest scientific contribution even considering his Nobel-prize winning work on QED. Such an elegant and amazing formalism is definitely worth further elaboration. This quantum version of the variational principle beautifully applies both geometry and algebra to physics in a united way. The geometry of physics and its physical structures could be described well in mathematical language of differential geometry. The structures are determined by underlying algebra, which vice versa defines the geometry.
Hans Bethe, another Nobel laureate, said, “There are two types of genius. Ordinary geniuses do great things, but they leave you room to believe that you could do the same if only you worked hard enough. Then there are magicians, and you can have no idea how they do it. Feynman was a magician.” Indeed, Feynman is certainly one of the most admired physicists in 20th century.
One of the most intriguing features of the variational principle is that every path or configuration contributes equally to the probability amplitude. This democracy principle is for the equal opportunities but definitely not for the equal results. Clearly, most of the contributions from various configurations cancel out each other in the resulting probability. Only the ones near an extremal action (that is maximally symmetric) are most probable.
The variational principle and associated math of differential geometry also ensures the linear nature of quantum mechanics as the differential operator is nilpotent or d^2=0. It states that quantumness is more fundamental in Mother Nature. Classical physics is emergent as contributions from most paths decohere and cancel out each other while only the most probable path survives. This leads to the principle of least action in classical physics.
Under the variational principle, the action S contains the essence of a physical model. How to construct S has been the main focus of physicists. The principle of finiteness becomes the so-called renormalizability in quantum theory. That is, all the terms used to construct S have to be renormalizable; otherwise, these infinities will cause the suppression of non-renormalizable terms in S due to the exponential phase factor. Asymmetries in S also cancel out each other resulting in a maximally symmetric action for a given model. As for what to be used or the physical contents in constructing S, we have to resort to the spacetime principle.
A well-known verse in “Tao Te Ching”[道德经] by Lao Tzu[老子] says that “Tao produced one; one produced two; two produced three; three produced all things” [道生一，一生二，二生三，三生万物。]. It amazingly depicts the big bang dynamics of the early Universe and dimensional phase transitions of spacetime. Here Tao could be understood as quantum chaos without spacetime. One means the one time dimension that was first inflated in our Universe (under SMM1 of the new theory). Two means the two-dimensional spacetime governed by SMM2. Maybe we should replace three with four here. That is, the last step is the fully inflated four-dimensional spacetime that holds everything we have observed in the current Universe.
Now we can see how the Supersymmetric Mirror Models (SMM1, SMM2, and SMM4) were constructed under the new mirror matter theory for varied spacetime dimensions using these first principles.
For an n-dimensional spacetime geometry (or Riemannian manifold, to be exact), the maximum symmetry group for metric preservation (in other words, making measurements meaningful) or the holonomy symmetry is the orthogonal group O(n)=O(1)*SO(n). This symmetry group is why one can make rationalized measurements at different spacetime points. It always has a simple discrete subgroup O(1) = Z2 that contains two elements of {1,-1}. We call this Z2 group the mirror symmetry that describes the orientation of spacetime geometry.
In one-dimensional time manifold, the mirror symmetry is the same as the time reversal symmetry. The time dimension must emerge first in the geometry so that the Universe can have its first symmetry — time reversal symmetry spontaneously broken and become dynamic and causal afterwards. The underlying model is SMM1 or 1-d Supersymmetric Mirror Model. In 1-d geometry, there is no intrinsic curvature, that is, gravity does not exist yet. There is no continuous gauge symmetry and no supersymmetry, either. That is, no gauge interactions and no spinor particles exist except for a simple scalar field. The reason why the model is still called SMM1 is because it is the simplest 1-d extension to other SMM models in higher dimensions.
The only particle field that can exist in SMM1 is a scalar that can obtain mass via a Higgs-like mechanism. This will spontaneously break the time reversal symmetry under the symmetry breaking model SMM1b. The time arrow then appears due to the breakdown of the Z2 time reversal symmetry. The first order of the Universe is then established: the time order or causality. Indeed, the time reversal symmetry is the first emergent symmetry and also first broken under the new theory.
The resulting heavy scalar then leads to the inflation of one space dimension. After the scalar decays into the next generation of particles, the 2-d spacetime is governed by SMM2. The holonomy group then becomes O(2) = Z2 * U(1). Both U(1) gauge bosons and Majorana fermions (a particle is also its own antiparticle) are born in this phase due to the emergence of supersymmetry. Fermions here have to be of Majorana type instead of Dirac type because antiparticle’ degrees of freedom can not be set free in 2-d spacetime. The Z2 mirror symmetry is equivalent of the chiral symmetry in this case. The two left- and right-handed sectors are completely decoupled. Gravity first appears in 2-d form which can also be described by a 2-d conformal field theory. As it turns out, the interior of a black hole is also 2-d in nature governed by SMM2.
At the next stage, condensation of Majorana fermions creates two heavy scalars (described under SMM2b) that spontaneously break the symmetries again leading to the inflation of two additional space dimensions. The heavy scalars then decay into the new generation of particles in 4-d spacetime under SMM4. Under the new holonomy group, i.e., the so-called Lorentz group O(4) = Z2 * Z2′ * SO+(1,3), the quantum counterpart contains also two copies of Z2 groups where one is the mirror or orientation symmetry and the other is the time reversal or equivalently CP symmetry. The restricted SO+(1,3) group requires the emergence of U(1) and SU(2) gauge groups and three generations of leptons as Dirac fermions (antiparticles now possess separate degrees of freedom) corresponding to the induced local / tangent / fiber space. Supersymmetry in 4-d spacetime, in addition, demands the emergence of 6 additional dimensions of compactified or tightly curved space (i.e., Calabi-Yau space). This 6-d Calabi-Yau space results in quarks and their SU(3) gauge interactions and consequently the balance of degrees of freedom between gauge bosons and matter fermions for supersymmetry. The compactification of the 6-d space also causes quark confinement. Due to the mirror symmetry, there are also another copy of similar particles and gauge interactions in the mirror sector. The two sectors are completely decoupled in terms of gauge interactions although both sectors share the same extended 4-d spacetime geometry or gravity that is described by Einstein’s general relativity.
Then, staged quark condensation leads to spontaneous symmetry breaking again and provides masses to the particles (SMM4b for the current Universe). In particular, neutrinos become degenerate and gain tiny masses due to a very small mass splitting of the two sectors. Therefore, the ordinary sector becomes left-handed while the mirror one right-handed to preserve renormalizability or pseudo-supersymmetry. This gives the well-known Standard Model of particle physics for the ordinary sector and similarly for the mirror sector. The two sectors of ordinary and mirror worlds are connected via degenerate neutrinos after the symmetry breaking, in particular, the mass terms of neutrinos that involve both left-handed ones in our sector and right-handed ones in the mirror sector.
The small mass splitting of the two sectors can explain the observed scale of dark energy [arXiv:1908.11838] and is also responsible for ordinary-mirror neutral hadron oscillations [arXiv:1906.10262][Phys. Lett. B 797, 134921 (2019)]. In particular, oscillations of neutrons (n-n’) and kaon mesons (K0-K0′) turn out to be the two most important messengers between the two sectors. The two oscillations provide a consistent picture for the origin of both dark matter and matter-antimatter imbalance in the Universe [Phys. Rev. D 100, 063537 (2019)]. In addition, n-n’ oscillations can also explain a lot of other puzzles: ultrahigh energy cosmic rays, synthesis of heavy elements and stellar evolution, neutron lifetime anomaly, etc. More intriguingly, feasible laboratory experiments with unique predictions are proposed to test the new n-n’ model and could uncover further evidence of the new physics [arXiv:1906.10262].
For example, each incoherent scattering or interaction of a neutral hadron in ordinary matter medium will result in a loss to its mirror counterpart. In terms of neutrons, this loss rate due to n-n’ oscillations is about 10-5 per bounce (or 0.45-1 × 10-5 as constrained in [arXiv:2006.10746]). This can perfectly explain the so-called neutron lifetime anomaly. It could also be used to further test the new theory in neutron lifetime measurements with magnetic traps or super-strong magnetic fields.
Under this new theory, one can see that gravity emerges as smooth spacetime geometry inflates dimensionally at D≥2. That is, gravity is truly a classical phenomenon, and also possibly the progenitor of all classical phenomena. Quantum measurement problem is probably related to classical extended spacetime. A measurement can cause a quantum wave function collapsing to its individual eigenstate since the measurement has to be conducted under the metric of the spacetime manifold, which is classical by definition.
The holonomy symmetry group including the mirror symmetry is the critical symmetry group originated in the spacetime inflation. The mirror symmetry corresponds to the orientation symmetry of the spacetime manifold. All other continuous gauge symmetries also come from the local / tangent / fiber space (including compactified space) that is derived from the base manifold of spacetime with renormalizability and supersymmetry. For a given spacetime geometry, particle fields and gauge interactions are therefore well-defined. Supersymmetry is a natural requirement due to finiteness or renormalizability for spacetime with two or more dimensions. It essentially transforms between matter fermions and gauge bosons. However, it is the mirror symmetry instead of supersymmetry that introduces another mirrored copy of particles making up the so-called dark matter of the Universe.
As a metric-preserving symmetry, the holonomy group inevitably involves deeply the concept of loops (closed paths on the manifold). This is possibly why string theory and loop quantum gravity can be good math or theoretical tools in studying the proposed new framework under certain scenarios. The algebra SO(2)=U(1) may also indicate why quantum theory in D≥2 spacetime becomes complex. As such, all these concepts of holonomy, loop, string, and complex are probably profoundly related to each other for D≥2 spacetime.
In short summary, principle #1 leads to the holonomy group for consistency and renormalizability (including supersymmetry) for finiteness; principle #2 defines the differential geometry of the quantum theory (linear, democratic, and quantum); principle #3 introduces the dynamics of the spacetime base manifold. The classical spacetime and its quantum contents can then emerge naturally in the Universe. In terms of mathematical language of fiber bundles, quantum physics or the fiber space is solely determined by the given classical spacetime or the base manifold under the above first principles.
As seen above, all physical phenomena are emergent as the Universe unfolds. All observed symmetries are emergent. All particles and laws are emergent. And in a dynamic way – a series of dimensional phase transitions in spacetime. As shown in Einstein’s GR theory, gravity, as a pure classical phenomenon, is about the smoothly inflated spacetime geometry. Quantum theory, on the other hand, describes the non-smooth and discrete part of the world.
What about the emergence of spacetime itself? For a distant or classical picture, it seems to be fine to use a continuous math framework or smoothly continuous manifold to describe the evolution of spacetime. But upon a close examination, in particular, around phase transitions, a discrete picture of spacetime may be needed. There are several candidates of theoretical tools for such a picture: loop quantum gravity, causal dynamical triangulation, causal sets.
The critical point is discreteness or quantumness. But how the quantumness is organized? There are several descriptive words that possibly tell the same truth or different aspects of the same truth: random, anarchic, democratic, chaotic, self-similar, collective, fractal, universality, cellular automaton, etc. This might be true for the first emergence of time dimension and that of each individual space dimension. We clearly need better tools to reveal the intriguing details of such dimensional phase transitions in spacetime.
