.. not the mass gap problem of Yang-Mills theories

The purpose of this tab is to put the relevant terms and concepts of gauge theories into the context of the paradigm changes enabled by a Krein space framework. The related affected notions and mathematical concepts of the new paradigms are summarized in the tab "Quanta systems scheme".

 
"There isn’t any theory that has U(1) x SU(2) x SU(3)"
Quote from R. Feynman, (GlJ) p. 433

The standard model is that the one that says that we have electrodynamics, we have weak interaction, and we have strong interaction; .. They’re not put together. … Three theories, strong interactions, weak interactions, and electromagnetic …  The theories are linked because they seem to have similar characteristics … Where does it go together? Only if you add some stuff that we don’t know. There isn’t any theory today that has SU(3) x SU(2) x U(1).

(GlJ) Gleick J., Genius, The life and science of Richard Feynman, Vintage Books, New York, 1991


Report on the status of the Yang-Mills problem
Remarks from (DoM)

As explained in the official CMI problem description set out by Arthur Jaffe and Edward Witten, Yang- Mills theory is a generalization of Maxwell’s theory of electromagnetism, in which the basic dynamical variable is a connection on a G-bundle over four-dimensional space-time. Its quantum version is the key ingredient in the Standard Model of the elementary particles and their interactions, and a solution to this problem would both put this theory on a firm mathematical footing and demonstrate a key feature of the physics of strong interactions.

(DoM) Douglas M. R., Report on the status of the Yang-Mills problem  


The standard model is not the final theory
Remark from (SaR) p. 82

Theoretical physicists are convinced that the standard model is not the final theory. There are a number of phenomena which find no explanation in the context of the standard model and must be added in an ad hoc manner. For example, the Higgs mechanism, the mysterious field which gives mass to all other particles, does not follow in any sense from the standard model. The apparent asymmetry between matter and anti-matter is not explained by the standard model. Neutrino masses do not naturally arise in the context of the standard model. There is clearly physics, a deeper theory, beyond the standard model.

(SaR) Sanders R. H., The Dark Matter Problem, A Historical Perspective, Cambridge University Press, Cambridge, 2013


Quantization of spinor fields, symmetries in the functional formalism
Remarks from (PeM)

(PeM) p. 306: We have now seens that the quantum field theoretic correlation functions of scalar, vector, and spinor fields can be computed from the functional integral, completely bypassing the construction of the Hamiltonian, the Hilbert space of states, and the equations of motion. The functional integral formalism makes the symmetries of the problem manifest; any invariance of the Lagrangian will be an invariance of the quantum dynamics. However, we would like to be able to appeal also to the conservation laws that follows from the quantum equations of motion, or to these equations of motion themselves. …. We will see that the functional integral gives, in a most direct way, a quantum generalization of Noether’s theorem.


(PeM) Peskin M. E., Schroeder D. V., An Introduction To Quantum Field Theory, Westview Press, 2016


Gauge theories with spontaneous symmetry breaking
Remarks from (PeM)   

(PeM) p. 347: Spontaneously broken symmetry is a central concept in the study of quantum field theory, for two reasons. Firs, it plays a major role in the applications of quantum field theory to Nature. In this book, we will see two very different examples of such applications: Chapter 13 will apply the theory of hidden symmetry to statistical mechanics, specifically to the behavior of thermodynamic variables near second-order phase transitions. Later, in Chapter 20, we will se the hidden symmetry is an essential ingredient in the theory of th weak interactions. Spontaneous symmetry breaking also finds applications in the theory of the strong interactions, and in the search for unified models of fundamental physics, p. 347.

(PeM) p. 689: In the course of this book, we have discussed three distinct fashions in which symmetries can be realized in a quantum field theory. The simplest case is a global symmetry that is manifest, leading to particle multiplets with restricted interactions. A second possibility is a global symmetry that is spontaneously broken. Then, as discussed in Chapter 11, the symmetry currents are still conserved and interactions are similarly restricted, but the vacuum state does not respect the symmetry and the particles do not form obvious symmetry multiplets. Instead, such a theory contains massless particles, Goldstone bosons, one for each generator of the spontaneously broken symmetry. The third case is that of a local, or gauge, symmetry. As we saw in Chapter 5, such a symmetry requires the existence of a massless vector field for each symmetry generator, and the interactions among these fields are highly restricted.   It is now only natural to consider a fourth possibility: What happens if we include both local gauge invariance and spontaneous symmetry breaking in the same theory? In this chapter and the next, we will find that this combination of ingredients leads to new possibilities for the construction of quantum field theory models. We will see that spontaneous symmetry breaking requires gauge vector bosons to aquire mass. However, the interactions of these massive bosons are still constrained by the underlying gauge symmetry, and these constraints can have observable consequences.

(PeM) Peskin M. E., Schroeder D. V., An Introduction To Quantum Field Theory, Westview Press, 2016


Open problems: Yang-Mills existence and mass gap
Notes from (ChS)

Open problem #1: Yang-Mills existence

The most interesting gauge groups are non-Abelian Lie groups like SU(2) and SU(3). It is not clear what the scaling limit should look like, or what space it should belong to. Even if one is able to somehow obtain a scaling limit, it is important to prove that it is nontrivial | meaning that it is a non-Gaussian field (on whatever space it's defined on).

Open problem #2: Mass gap

Various Yang-Mills theories (such as 4D Yang-Mills theory with gauge group SU(3)) are supposed to have mass gaps. At small „beta“, exponential decay can be proved by well-known techniques from statistical physics (expansions or coupling). No general method for large „beta“.


(ChS) Chatterjee S., Yang-Mills for mathematicians, Columbia University


The self-energy of a free electron
Note from AI

The self-energy of a free electron is the energy of the electromagnetic field it generates and interacts with, a concept that initially led to infinities in early Quantum Electrodynamics (QED). This problem was resolved by renormalization, a technique that effectively absorbs the infinite parts into the observed mass and charge of the electron, making the theory consistent. The self-energy also refers to the energy shift an electron experiences due to its interaction with its environment, such as confined spaces or vacuum fluctuations, leading to observable effects like changes in energy levels or relaxation rates.


The real and complex Lorentz groups
Remarks from (StR) pp. 9 ff.

Among the most important symmetries of relativistics quantum theory are those which arise from the Lorentz transformations themselves. … The Lorentz transformations form a group, the Lorentz group L. … It makes sense to say that two Lorentz transformations can be connected to one another by a continuous curve of Lorentz transformations. L has four components, each of which is connected in the sense that any one point can be connected to any other, but no Lorentz transformation in one component can be connectd to another in another component. The four components allow the building of three important sub-groups of the Lorentz group L, (1) the orthochronous Lorentz group, (2) the proper Lorentz group, and (3) the orthochorous Lorentz group. The restricted Lorentz group is the common sub-group of those three sub-groups. It excludes those elements of the full Lorentz group, which include reversals of space-time directions. It is therefore also called the “core” group of the Lorentz group and the above three Lorentz sub-groups.

The restricted Lorentz group is associated with the group of 2x2 complex matrices of determinant one, which is denoted by SL(2,C) (S stands for special meaning determinant one, L for linear, 2 is the dimension, and C stands for complex), which is isometric to the unit quaternions S(3). The restricted Lorentz group is important in describing the transformation properties of spinors. In SMEP the weak force interaction of the Quantum Flavordynamics (QFD) enabled by the three (boson-mediator) "particles" W(+), W(-), Z, is characterized by the restricted Lorentz (symmetry) group.

Another group associated with the Lorentz group, L, is the complex Lorentz group L(C). It is essential to prove the PCT theorem. L has two connected components. Just as the restricted Lorentz group is associated with SL(2,C) the complex Lorentz group is associated with SL(2,C) x SL(2,C). This latter group is the set of all pairs of 2x2 matrices of determinant one accompanied by a corresponding the multiplication law.


(StR) Streater R. F., Wightman A. S., PCT, Spin & Statistics, and all that, W. A. Benjamin, Inc., New York, Amsterdam, 1964


Pauli's spin operators and Hamilton's quaternions

Quote from R. Feynman: "Pauli's spin matrices and operators were nothing but Hamilton's quaternions", (UnA2) p.153.

Quote from A. Unzicker: "All in all, there are many indications that electrons, including their strange spin behavior, are described more simple by S(3). In any case, despite the elegant representation Dirac had developed, it cannot be claimed that this sheds light on the reason for the existence of spin", (UnA2) p. 183.


(UnA2) Unzicker A., The Mathematical Reality, Why Space and Time are an Illusion, 2020


The quaternion rotation operator
Remarks from (KuJ)

The quaternions provide an appropriate field to address the „translation-rotation“ (linear and angular rotation) „permutation“ requirement. The perhaps primary application of quaternions is the quaternion rotation operator. This is a special quaternion triple-product (unit quaternions and rotating imaginary vector) competing with the conventional (Euler) matrix rotation operator. The quaternion rotation operator can be interpreted as a frame or a point-set rotation, (KuJ). 

Its outstanding advantages compared to the Euler geometry are

- the axes of rotation and angles of rotation are independent from the underlying coordinate system and direct readable

- there is no need to to take care about the sequencing of the rotary axes.


Conservation laws, invariance, and symmetry
Remarks from (NeD) pp. 1-4

Conservation laws are a fundamental principle of physics. E. Noether’s theorem relates classes of conservation laws to symmetries of space, time, and “internal” variables. To make symmetry quantitative, one needs to carry out some operation, or transformation, and see if one can tell the difference. If the difference is precisely zero, or – more operationally relevant – if the difference is too small to detect within some infinitesimal tolerence, then the system is called “invariant under the transformation. …. “. Invariant” means that a quantity’s numerical value is not altered by a coordinate transformation. “Conserved” in contrast, means that within a given coordinate system the quantity does not change throughout a process. “Invariance” compares a quantity between reference frames. “Conservation” compares the quantity before and after collision or reaction or process within a reference frame. Noether’s theorem relates conservation to invariance, and thus to symmetry.


Assumptions of relativistic quantum theory
Remarks from (StR) pp. 96, 97

The classical notion of field originated in attempts to avoid the idea of action at a distance in the prescription of electromagnetics and gravitational phenomena. In these important cases, the field turns out to have two properties: (1) it is observable, (2) it is defined by a set of functions on space-time with well-defined transformation law under the appropriate relativity group. Since in quantum mechanics observables are represented by hermitian operators which act on the Hilbert space of state vectors, one expects the analogue in relativistic quantum mechanics of a classical observable field to be a set of hermitian operators defined at each point od space-time and having a well-defined transformation law under the appropriate group. ... It turns out that not only observable but also unobservable fields are of interest, so we shall also consider non-hermitian operators.


Bohm's notions of undivided wholeness, implicate and exlicate orders
Remarks from (BoD1)

What suggests itself, then, is that we seriously consider the possibility of dropping the idea of the basic role of signal, but go on with the other aspects of relativity theory (especially the principle that laws are invariant relationships, and that through non-linearity of the equations, or in some other way, analysis into autonomous components wlll cease to be relevant). Thus, by letting go of this kind of attachment to a certain kind of analysis that does not harmonize with the “quantum” context, we open the way for a new theory that comprehends what is still valid in relativity theory, but does not deny the indivisble wholeness implied by the quantum theory, p. 173. 

To begin with undivided wholeness means, that we must drop the mechanistic order. But this order has been, for many centuries, basic to all thinking on physics. As brought out in chapter 5, the mechanistic order is most naturally and directly expressed through the Cartesian grid. .. To help make it easier to see what is meant by our proposal of new notions of order that are appropriate to undived wholeness, it is therefore useful to start with examples that may directly involve sense perception, as well as with models and analogies that illustrate such notions in an imaginative and intuitive way. In chapter 6 we began by noting the photographic lens is an instrument that has given us a very direct kind of sense perception of the meaning of the mechanistic image, it very strongly calls attension to the separate elements into which the imaging and recording of things that are too small to be seen with naked eye, too big, too fast, too slow, etc., it leds us to believe that eventually everything can be perceived in this way. From this grows the idea that there is nothing that cannot also be conceived as constituted of such localized elements. Thus the mechanistic approach was greatly encouraged by the development of the photographic lens.

We then went on to consider a new instrument, the hologram. .. The key new feature of this record is that each part contains information about the whole object (so that there is no point-to-point correspondence of object and recorded image). That is to say, the form and structure of the entire object may be said to be enfolded within each region of the photographic record. When one shines light on any region, this form and structure are then unfolded to give a recognizable image of the whole object once again. We propose that a new notion of order is involved here, which we called the implicate order. In terms of the implicate order one may say that everythnig is enfolded into everything. This contrasts with the explicate order now dominant in physics in which things are unfolded in the sense that each thing lies only in its own particular region of space (and time) and outside the regions belonging to other things, pp. 223, 224.