GAUGE FIELDS, PHYSICAL REALITY AND MATHEMATICAL REALITY

Physical reality is that which can be detected with any measuring device that can be physically constructed. However, in the configuration of what we call "reality," there are also magnitudes that cannot be measured directly . These magnitudes seem to lie halfway between the real physical world and the "idealized" world of mathematics.

In this article, we will see how the relationship between these "hidden" quantities and the so-called gauge symmetry leads us to a different reality: a reality that only manifests when we make a measurement, that is, when we establish a reference system relative to which we perform the measurements. We will also see how these "hidden" quantities reveal information that initially seems completely inaccessible, information that is nonlocal outside our causal zone.

Welcome to the hidden side of reality!

Gauge symmetry

The Lagrangian of a physical system is a fundamental quantity that allows us to describe the dynamics of that system. Broadly speaking, the Lagrangian can be defined as the difference between kinetic energy and potential energy. Suppose a system of N particles is defined by the Lagrangian:

This Lagrangian only depends on the derivative of x with respect to t, so including any constant in the Lagrangian will not change the result. This means that any Lagrangian of type L+cte will produce the same results as the Lagrangian L and therefore represent the same physical system .

This is the meaning of gauge symmetry: if we modify the Lagrangian of a system so that it does not change its value then this modified Lagrangian describes the same physical system . Since we have many equivalent Lagrangians that actually describe the same physical system, the gauge symmetry represents a redundancy of the system (although as we will see in this article, there is something deeper than simple redundancy). The existence of these redundant systems leads us to our fundamental gauge principle: only quantities that remain invariant under gauge changes have physical meaning , these quantities are called gauge invariants.

This fundamental principle has led 20th-century physicists to one of the greatest achievements of human knowledge: the Standard Model of particle physics, which explains all the fundamental forces of nature except gravity.

Gauge symmetry and electromagnetic interaction

Virtually all technology in our world is based on electromagnetic interaction. This interaction is made possible by U(1) gauge symmetry. To understand this, we'll explain in a simple way the process of interaction between a charged particle and the electromagnetic field.

The Lagrangian for a free particle of mass m is:

Where:

and the term "gamma" represents the famous Pauli matrices that satisfy:

This Lagrangian has a global U(1) gauge symmetry since if we change the phase of the Lagrangian wave function by any angle, the Lagrangian remains unchanged. This is because, as we know from the foundations of quantum mechanics, physical quantities such as the probability of finding the particle at a specific point in space-time are given by the square of the modulus of the wave function. In fact, we know that only the modulus has physical meaning; therefore, the phase is not physically measurable (1).

The wave function of a particle is a complex function. Every complex number can be represented as a vector consisting of a magnitude and an angle. U(1) symmetry consists of changes in the phase (the angle) of the complex wave function.

Noether's theorem states that every global symmetry implies a conserved charge. Therefore, the total charge associated with the U(1) gauge symmetry must be conserved.

Our free particle is completely isolated from the outside world. For all intents and purposes, it's as if our particle lived in an empty Universe with no reference frames. In this empty Universe, there is no way to establish any physical measurements, since positions and momenta must be measured relative to another physical system . To describe the dynamics of any physical particle, we must localize the particle in space-time. This implies that the phase of the particle's wave function must vary depending on its motion in space-time. This is equivalent to performing a local transformation, that is, a transformation in which the phase angle depends on the space-time coordinates.

Next we are going to "localize" our particle, that is, we are going to apply a local transformation to our free particle which simply consists of changing the phase of the wave function by an angle theta(x):

By applying this transformation to the Lagrangian and calculating the derivative with respect to theta(x) we obtain:

The result is that we have obtained our original Lagrangian, minus one extra term. This extra term takes the form of an interaction where Q is the value of the charge (the coupling). This means that localizing a particle is equivalent to performing an interaction with the particle (logically, any physical action inevitably implies an interaction with the measured system; in this case, the particle is localized with respect to a specific point in the electromagnetic field). By performing this transformation, we have broken the global U(1) gauge symmetry, since our Lagrangian is now not invariant under phase changes. In addition, we have "added" an amount of charge Q to the Lagrangian. However, the existence of a conserved global charge implies that there must be a mechanism to counteract this new term and its associated charge. One way to reestablish the global symmetry would be to take the derivative of the angle and obtain a term that has the following form:

Where A is a field with U(1) symmetry that remains invariant under changes in the complex phase. The new derivative we have defined is called the covariant derivative. Using it in our original Lagrangian, we obtain:

If we then perform the local gauge transformation again with the new covariant derivative, the symmetry is restored! However, we now obtain an interaction with a new field A. What does this new field mean? This field is called the gauge field , and it can be easily shown that the variation of this field produces the force field of the electromagnetic field F:

That is, F=dA. This means that the restoration of the global gauge symmetry and the conservation of its associated charge implies electromagnetic interaction !

Initially, the electromagnetic field and the particle are isolated. There is global gauge symmetry; there are no local reference frames.

The particle is then "localized." This leads to an interaction, the breaking of the overall gauge symmetry, and the appearance of an extra charge.

Finally, the conservation of total charge implies the restoration of overall symmetry with the emergence of an interaction that counteracts the extra charge. This interaction is the electromagnetic force.

The importance of this "new" field is enormous: it is responsible for all the technology that exists in our world. However, the gauge field A is not directly measurable; in fact, we can only measure its invariant gauge components E and B (the electric field and the magnetic field). How is it possible that the fundamental field from which all the laws of electromagnetic interaction are derived is an undetectable field? If this field is merely redundant, how is it possible that all these fundamental laws are derived from it?

Although it is possible to derive the electromagnetic force field without reference to the gauge field, there are phenomena such as the Aharonov-Bohm effect that are impossible to explain without reference to the gauge field potentials. Furthermore, while the Lagrangian of a free particle is gauge-independent (dA=0), we have already seen that the Lagrangian of a particle interacting with the electromagnetic field is:

This means that it is not gauge invariant since it couples to the field A and not to the gauge invariant components E and B. This seems to indicate that the gauge field is more than a mere redundancy (2): this field contains the possible values with which it is possible to couple the particle to the electromagnetic field so that when an interaction occurs one of the possible values is selected: the value that we physically measure.

Interactions, intertwining, global reality and local reality

There is a simple expression from which all of Quantum Mechanics can be derived: [q,p]=ih. This expression tells us that the quantities q and p that represent the position and momentum of a particle do not have an independent existence, that is, the usual Cartesian product qxp that consists of multiplying the states of q by the states of p is not valid in this case (that is why in quantum mechanics we need to use the tensor product). In quantum mechanics, by multiplying the states of q by the states of p we find that there are more states than initially there were in q and p. These new states are interaction states, more specifically, they are entanglement states between q and p. The fundamental conclusion is the following: certain quantities such as the position and momentum of a particle do not have an independent existence but only exist in a superimposed or entangled form. Furthermore, this existence only manifests when a measurement is made, that is why whenever we make a measurement we obtain an extra term: the interaction term related to the gauge symmetry. This is why gauge symmetry is so ubiquitous in fundamental physics: it represents an interaction term and physical reality only manifests itself when there are interactions (3).

The existence of these entangled magnitudes has a fascinating consequence: although by the principle of relativity no interaction can propagate faster than light, we can make statistical measurements of correlated magnitudes and thus obtain non-local information that is outside our causal reality . The best-known example of this is the entanglement between two particles created with opposite momenta: by measuring the spin of one of them we automatically know the spin of the other even if they are light years away. This is why it is said that quantum mechanics predicts the existence of non-local effects. Another way to see this is through Feynman path integrals: in this formulation of quantum mechanics, we must consider all possible paths between two points and calculate the interference between the complex amplitudes of these paths such that the paths with opposite phase cancel each other out. This tells us that although phase is not measurable locally by measuring a single particle, it can be detected statistically by measuring correlated or entangled variables. Physicists can use these effects to study global characteristics of our Universe, such as its global topology. Once again, modern physics, armed with the power of mathematics, allows us to reach places that seem impossible to explore.

Physical Reality and Mathematical Reality

The physicist and philosopher Ernest Mach defended the relative nature of physics: every measured quantity must be defined relative to a reference system; that is, there are no absolute quantities; every quantity is defined relative to "something else" (a physical reference system). Einstein's admiration for Mach was such that one of his fundamental goals was to try to introduce Mach's principle into General Relativity. More than a century later, modern physics seems to support Mach's view.

Gauge symmetry tells us that in our fundamental physical theories there is an "ambiguity" or redundancy: many different Lagrangians actually represent the same physical system. This ambiguity disappears as soon as we establish a reference frame, that is, when we perform a measurement. The measurement involves an interaction term between two systems. Gauge symmetry dictates how these systems can couple, so its role in shaping reality appears to be deeper than that of simple redundancy.

Our modern theories of quantum gravity seem to indicate that quantum entanglement is linked to the connectivity of space-time, implying a quantum duality between entanglement and gravity (ER=EPR duality). All of this leads us to a Universe connected through quantum entanglement, such that, although it is not possible to send information superluminally, we can detect certain nonlocal phenomena. In fact, the most convincing explanation for certain fundamental phenomena, such as the information paradox in black holes, comes from the existence of nonlocal phenomena such as wormholes (see this article ).

Once again, modern physics shows us that our naive, everyday view of the world around us is only a crude approximation of true reality: an astonishing reality that defies our ability to visualize.

Grades:

(1) As also explained in the article, although the phase is not measurable in a single particle, it can be detected statistically by measuring entangled magnitudes.

(2) There is some controversy about the true meaning of gauge symmetry. The version presented in this article is based on Carlo Rovelli's article "Why Gauge?" mentioned in the "Sources" section.

(3) This interpretation is aligned with the view of relational physics set forth in the article "Why Gauge?" cited at the end of this article.

Sources:

Why gauge? , The hidden quantum origin of gauge connections , Gauge Symmetry in QED