I am a mathematics student with a hobby interest in physics. This means that I've taken graduate courses in quantum dynamics and general relativity without the bulk of undergraduate physics courses and sheer volume of education into the physical tools and mindset that the other students who took the course had, like Noether's theorem, Lagrangian and Hamiltonian mechanics, statistical methods, and so on.
The courses themselves went well enough. My mathematical experience more or less made up for a lacking physical understanding. However, I still haven't found an elementary explanation of gauge invariance (if there is such a thing). I am aware of some examples, like how the magnetic potential is unique only up to a (time-)constant gradient. I also came across it in linearised general relativity, where there are several different perturbations to the spacetime metric that give the same observable dynamics.
However, to really understand what's going on, I like to have simpler examples. Unfortunately, I haven't been able to find any. I guess, since "gauge invariance" is such a frightening phrase, no one use that word when writing to a high school student.
So, my (very simple) question is: In many high school physics calculations, you measure or calculate time, distance, potential energy, temperature, and other quantities. These calculations very often depend only on the difference between two values, not the concrete values themselves. You are therefore free to choose a zero to your liking. Is this an example of gauge invariance in the same sense as the graduate examples above? Or are these two different concepts?
Answer
The reason that it's so hard to understand what physicists mean when they talk about "gauge freedom" is that there are at least four inequivalent definitions that I've seen used:
Definition 1: A mathematical theory has a gauge freedom if some of the mathematical degrees of freedom are "redundant" in the sense that two different mathematical expressions describe the exact same physical system. Then the redundant (or "gauge dependent") degrees of freedom are "unphysical" in the sense that no possible experiment could uniquely determine their values, even in principle. One famous example is the overall phase of a quantum state - it's completely unmeasurable and two vectors in Hilbert space that differ only by an overall phase describe the exact same state. Another example, as you mentioned, is any kind of potential which must be differentiated to yield a physical quantity - for example, a potential energy function. (Although some of your other examples, like temperature, are not examples of gauge-dependent quantities, because there is a well-defined physical sense of zero temperature.)
For physical systems that are described by mathematical structures with a gauge freedom, the best way to mathematically define a specific physical configuration is as an equivalence class of gauge-dependent functions which differ only in their gauge degrees of freedom. For example, in quantum mechanics, a physical state isn't actually described by a single vector in Hilbert space, but rather by an equivalence class of vectors that differ by an overall scalar multiple. Or more simply, by a line of vectors in Hilbert space. (If you want to get fancy, the space of physical states is called a "projective Hilbert space," which is the set of lines in Hilbert space, or more precisely a version of the Hilbert space in which vectors are identified if they are proportional to each other.) I suppose you could also define "physical potential energies" as sets of potential energy functions that differ only by an additive constant, although in practice that's kind of overkill. These equivalence classes remove the gauge freedom by construction, and so are "gauge invariant."
Sometimes (though not always) there's a simple mathematical operation that removes all the redundant degrees of freedom while preserving all the physical ones. For example, given a potential energy, one can take the gradient to yield a force field, which is directly measurable. And in the case of classical E&M, there are certain linear combinations of partial derivatives that reduce the potentials to directly measurable ${\bf E}$ and ${\bf B}$ fields without losing any physical information. However, in the case of a vector in a quantum Hilbert space, there's no simple derivative operation that removes the phase freedom without losing anything else.
Definition 2: The same as Definition 1, but with the additional requirement that the redundant degrees of freedom be local. What this means is that there exists some kind of mathematical operation that depends on an arbitrary smooth function $\lambda(x)$ on spacetime that leaves the physical degrees of freedom (i.e. the physically measurable quantities) invariant. The canonical example of course is that if you take any smooth function $\lambda(x)$, then adding $\partial_\mu \lambda(x)$ to the electromagnetic four-potential $A_\mu(x)$ leaves the physical quantities (the ${\bf E}$ and ${\bf B}$ fields) unchanged. (In field theory, the requirement that the "physical degrees of freedom" are unchanged is phrased as requiring that the Lagrangian density $\mathcal{L}[\varphi(x)]$ be unchanged, but other formulations are possible.) This definition is clearly much stricter - the examples given above in Definition 1 don't count under this definition - and most of the time when physicists talk about "gauge freedom" this is the definition they mean. In this case, instead of having just a few redundant/unphysical degrees of freedom (like the overall constant for your potential energy), you have a continuously infinite number. (To make matters even more confusing, some people use the phrase "global gauge symmetry" in the sense of Definition 1 to describe things like the global phase freedom of a quantum state, which would clearly be a contradiction in terms in the sense of Definition 2.)
It turns out that in order to deal with this in quantum field theory, you need to substantially change your approach to quantization (technically, you need to "gauge fix your path integral") in order to eliminate all the unphysical degrees of freedom. When people talk about "gauge invariant" quantities under this definition, in practice they usually mean the directly physically measurable derivatives, like the electromagnetic tensor $F_{\mu \nu}$, that remain unchanged ("invariant") under any gauge transformation. But technically, there are other gauge-invariant quantities as well, e.g. a uniform quantum superposition of $A_\mu(x) + \partial_\mu \lambda(x)$ over all possible $\lambda(x)$ for some particular $A_\mu(x).$
See Terry Tao's blog post for a great explanation of this second sense of gauge symmetry from a more mathematical perspective.
Definition 3: A Lagrangian is sometimes said to posses a "gauge symmetry" if there exists some operation that depends on an arbitrary continuous function on spacetime that leaves it invariant, even if the degrees of freedom being changed are physically measurable.
Definition 4: For a "lattice gauge theory" defined on local lattice Hamiltonians, there exists an operator supported on each lattice site that commutes with the Hamiltonian. In some cases, this operator corresponds to a physically measurable quantity.
The cases of Definitions 3 and 4 are a bit conceptually subtle so I won't go into them here - I can address them in a follow-up question if anyone's interested.
Update: I've written follow-up answers regarding whether there's any sense in which the gauge degrees of freedom can be physically measurable in the Hamiltonian case and the Lagrangian case.
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