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Symmetry → connection field → force of nature

Here’s my take on gauge symmetry after reading about the concept (some references are listed below).

There are mathematical frameworks for space, and in physics there are equations of motion for space. A mathematical framework deals with geometric and numeric objects. In everyday life we basically equate these objects with a Euclidean space using the Cartesian coordinate system. But there are other types of spaces and coordinate systems which serve better for many problems, particularly in physics where vector spaces are common.

Coordinate systems identify geometric or combinatorial objects with numerical (or standard) ones, but in many cases, there is no natural (or canonical) choice of this identification; instead, one may be faced with a variety of coordinate systems, all equally valid.  One can of course just fix one such system once and for all, in which case there is no real harm in thinking of the geometric and numeric objects as being equivalent.  If however one plans to change from one system to the next (or to avoid using such systems altogether), then it becomes important to carefully distinguish these two types of objects, to avoid confusion. [Ref 4]

Imagine you are texting with a sentient being from another galaxy and want to explain geometric forms. Like a right triangle. How would you do that so that the construct made sense no matter what the other’s frame of reference, coordinate system, geometry (Euclidean, non-Euclidian, etc.), or measurement scale? Your description would need to map to that other frame. [1]

In this context, you can’t draw (visualize) anything. And you’re not really interested in the size of the triangle, just that the shape and properties are clear. After explaining more basic abstract constructs like line segment, you then might be able to relate  ⎸AB2 + ⎸BC2 = ⎸AC2 to a right triangle. You’re defining a class of geometric shapes (not a particular instance) in common across different frames (and independent of unit of length).

There is no choice of “gauge” — no particular measurement metric (historically referring to railway gauge, a standard which allows mechanical “inter-connectivity and inter-operability”). Each local gauge, for example, can define a standard unit circle for the geometric class of circles [Ref 4].

No particular marking gauge either. Just an invariant geometric description (statements) that can be used to transfer a design or pattern to a medium without any particular unit of length — the transfer is “dimensionally consistent.” Then a local gauge ties down (fixes) the dimensions — breaks the invariant description — in order to actually build (manufacture) the subject of that design. But otherwise the design can be conveyed intact between local frames. The same pattern is mapped or “connected” to each.

In physics, breaking a local gauge invariant description by fixing a unit length, for example, helps simplify calculations.

For instance, in Euclidean geometry problems, it is often convenient to temporarily assign one key point to be the origin (thus spending translation invariance symmetry), then another, then switch back to a translation-invariant perspective, and so forth.  As long as one is correctly accounting for what symmetries are being spent and bought at any given time, this can be a very powerful way of simplifying one’s calculations.

But in many situations, it is not yet fully understood whether the use of the correct choice of gauge is a mere technical convenience, or is more innate to the equation.  It is definitely conceivable, for instance, that a given gauge field equation is well-posed with one choice of gauge but ill-posed with another.  It would also be desirable to have a more gauge-invariant theory of PDEs [partial differential equations] that did not rely so heavily on gauge theory at all, but this seems to be rather difficult; many of our most powerful tools in PDE (for instance, the Fourier transform) are highly non-gauge-invariant, which makes it very inconvenient to try to analyse these equations in a purely gauge-invariant setting. [Ref 4]

The math gets harder for connecting or transporting more complex properties across various geometries. For example, a gauge-invariant notion of derivative (as in differential equations). Exploiting the symmetries of space-time provides a consistent frame of reference.

Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. … Connections also lead to convenient formulations of geometric invariants, …

Physicists study natural laws that work the same in any frame, just like a shape in the above example. Take a set of particular equations and a local metric, do an experiment, and you’ll get the same results wherever you are (within experimental error). That’s (local) gauge symmetry.

So, how, as Sean Carroll says, do we go from local gauge symmetry to connection field to force of nature?

Since the experiments give the same result in different, independent locations (places in space and time), that tells us something about the character of that space-time. Imagine if the results were not the same, that everywhere and every time someone did that same experiment the results were different. Well, that would mean that the space-time conditions varied (at the particular emergent level in question). No law, eh. The fact that the experiments can be compared between locations (no matter what orientation, for example) tells us that there’s an implied commonality in the contexts, some connection which physicists like to call a field. In other words, if there wasn’t such a connection — a joint aspect, the results would vary. If there wasn’t a field, there’d be no natural law.

We take this for granted everyday with gravity, a field which locally alters how we move and the motion of objects around us — which we experience as forces on ourselves and the other objects.

That’s a gauge symmetry: when a symmetry transformation can be separately carried out at different points in space. Gauge symmetries are sometimes called local symmetries, since we can do them independently (locally) at every point; they are to be contrasted with global symmetries, which need to be done in a uniform way all over the place. It can be confusing, because “local” sounds like it’s less than “global,” whereas really a local/gauge symmetry represents enormously more symmetry than a mere global symmetry — infinitely more, since the transformations can happen completely independently at every point.

You might think of a gauge field as a latticework of invisible lines running through the universe, keeping track of what counts as “staying parallel” and “moving on a straight line” as we travel through space. But it’s a venerable principle of quantum field theory that, once you have a field, that field can have its own dynamics — it can bend and twist through space, typically in response to other fields that it interacts with. And when your gauge field starts twisting, you feel it as a force of nature.

What about the other forces — electromagnetism and the strong and weak nuclear forces? Nothing nearly so tangible, I’m afraid. These are all based on “internal” symmetries — they don’t transform things within space, but rather rotate different fields into each other. … Electromagnetism and the weak interactions follow a simple pattern. Gluons, photons, and W/Z bosons all arise from different kinds of connection fields relating the symmetry transformations at different points in space.[Ref 6]


[1] For example, as a systems engineer at Hughes, I used software routines to translate points of interest on the earth into satellite orbital coordinates and then to onboard equipment with its own axial coordinates. The process was done in reverse to convey information to the ground, with an appropriate frame of reference.



Dan Piponi, Studied theoretical physics at Cambridge Written Jan 24, 2015

For completeness I should point out some things that will allow you to compare with other people’s descriptions. The rule for how to compare fields at different points is known as a Connection (vector bundle). The rule for how to use a connection to get rates of change is called a Covariant derivative. The quantities that you’re comparing from one end of a path to another live in a space called a Vector bundle.


Charles H Martin, PhD & NSF Fellow, Theoretical Chemical Physics, University of Chicago Written Jul 11, 2011

In the context of physics and the Yang-Mills Gauge Theory…It is an idea ”that the fundamental symmetries of nature could actually dictate the character of the force fields of nature.”

3. Ibid…

Or, as Mills explained it in class to us…

“For every conservation law, there is a  symmetry. For every symmetry, there is a force field For every force field, there is a conservation law.”

4. What is a gauge? [long technical explanation]

27 September, 2008 in expository, math.AP, math.CO, math.DG, math.DS, math.MP | Tags: connections, curvature, fibre bundles, gauge fixing, gauge invariance, sections

… the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations).  By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed.  Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis.

5. Electromagnetic Field as a Connection in a Vector Bundle

Here’s a short(-ish) answer. A vector bundle is a family of vector spaces over a manifold. The vector spaces can have bases. The manifold can have coordinates. The two concepts are not a priori related (now for the bundle of tangent spaces, a coordinate change happens to induce a change of basis; this fact often sows confusion). Once you pick a basis for your vector space, you do define a vector by its components — but someone else may be describing the same vector in a different basis. To translate to physics: change of basis = gauge transformation. In the case of a charged particle, the wave function is the component of a one-vector section; in a new basis, this number changes by a non-zero complex number (which can vary from point to point). Again, the wave function is a section, and a section means one vector for each point in the manifold. How do we differentiate a function which takes values in different vector spaces over different points? We need a way of connecting the vector spaces. The connection does this; pragmatically, it is just a rule for doing this differentiation. Of course, differentiation will look different in different vector spaces, so the form of the connection will depend on the basis and change under gauge transformations (just as the form of a linear transformation changes under change of bases). That’s what various messy formulas about how things “transform” are trying to tell you.


3 thoughts on “Symmetry → connection field → force of nature

  1. A helpful Wiki article is Spontaneous symmetry breaking: “Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetrical state ends up in an asymmetrical state. In particular, it can describe systems where the equations of motion or the Lagrangian obey symmetries, but the lowest-energy vacuum solutions do not exhibit that same symmetry. … By definition, spontaneous symmetry breaking requires the existence of a symmetrical probability distribution — any pair of outcomes has the same probability. In other words, the underlying laws … are invariant under a symmetry transformation.”

    Graph of Goldstone's 'Mexican hat' potential function V versus φ.

    In particle physics, “An actual measurement reflects only one solution, representing a breakdown in the symmetry of the underlying theory. ‘Hidden‘ is a better term than ‘broken’, because the symmetry is always there in these equations. This phenomenon is called spontaneous symmetry breaking (SSB) because nothing (that we know of) breaks the symmetry in the equations.”

  2. Also helpful is the book Symmetries and Conservation Laws in Particle Physics by Stephen Haywood. A quite technical book. Below is a quote from an excerpt: Chapter 1 “Symmetries and Conservation Laws.”

    By definition, the system is said to have a symmetry if the Hamiltonian is invariant, i.e. H’ = H. Note: this is a symmetry of the Hamiltonian, not of the vector space (Hilbert space) of solutions {⍦}. It is H which defines the dynamics of the system, i.e. the interactions. Of course, the symmetry contained within H will be reflected in the individual solutions.

    To summarize: if the Hamiltonian of a system is invariant under a unitary transformation U generated by a Hermitian operator X, then there will be a conserved observable associated with X. Some of the transformations and their corresponding conserved observables in the case that the Hamiltonian is invariant are listed in Table 1.2.

  3. See also this American Journal of Physics (Vol. 72, No. 4, pp. 428–435, April 2004) article “Symmetries and conservation laws: Consequences of Noether’s theorem.”

    We derive conservation laws from symmetry operations using the principle of least action. These derivations, which are examples of Noether’s theorem, require only elementary calculus and are suitable for introductory physics. We extend these arguments to the transformation of coordinates due to uniform motion to show that a symmetry argument applies more elegantly to the Lorentz transformation than to the Galilean transformation.

    “It is increasingly clear that the symmetry group of nature is the deepest thing that we understand about nature today” (Steven Weinberg). Many of us have heard statements such as for each symmetry operation there is a corresponding conservation law. The conservation of momentum is related to the homogeneity of space. Invariance under translation in time means that the law of conservation of energy is valid. Such statements come from Noether’s theorem, one of the most amazing and useful theorems in physics. … Symmetries limit the possible forms of new physical laws. The deep connection between symmetry and conservation laws requires the existence of a minimum principle in nature: the principle of least action.

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