What is Gauge symmetry: Definition and 46 Discussions

In physics, a gauge theory is a type of field theory in which the Lagrangian does not change (is invariant) under local transformations from certain Lie groups.
The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, then the gauge theory is referred to as non-abelian gauge theory, the usual example being the Yang–Mills theory.
Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups. When they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. Local symmetry, the cornerstone of gauge theories, is a stronger constraint. In fact, a global symmetry is just a local symmetry whose group's parameters are fixed in spacetime (the same way a constant value can be understood as a function of a certain parameter, the output of which is always the same).
Gauge theories are important as the successful field theories explaining the dynamics of elementary particles. Quantum electrodynamics is an abelian gauge theory with the symmetry group U(1) and has one gauge field, the electromagnetic four-potential, with the photon being the gauge boson. The Standard Model is a non-abelian gauge theory with the symmetry group U(1) × SU(2) × SU(3) and has a total of twelve gauge bosons: the photon, three weak bosons and eight gluons.
Gauge theories are also important in explaining gravitation in the theory of general relativity. Its case is somewhat unusual in that the gauge field is a tensor, the Lanczos tensor. Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton. Gauge symmetries can be viewed as analogues of the principle of general covariance of general relativity in which the coordinate system can be chosen freely under arbitrary diffeomorphisms of spacetime. Both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of general covariance with a true gauge principle with new gauge fields.
Historically, these ideas were first stated in the context of classical electromagnetism and later in general relativity. However, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below. Today, gauge theories are useful in condensed matter, nuclear and high energy physics among other subfields.

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  1. D

    I Gauge and Phase Symmetries in Quantum Systems: Exploring the Confusion

    This is about the paper by Greiter: https://arxiv.org/pdf/cond-mat/0503400.pdf Greiter argues that local electromagnetic gauge symmetry cannot change the state of a quantum system. On the other hand, in QED charge or particle conservation (if energy is too low to produce particle-antiparticle...
  2. Pradyuman

    A How can gauge fields be associated with real particles?

    When we make our lagrangian invariant by U(1) symmetry we employ the fact that nature doesn't care how I describe it, but, how come that I can associate the real physical particles with the coordinates I use to describe? Even though gauge symmetry is not a physical Symmetry,
  3. Baela

    A Basic Question about Gauge Transformations

    Suppose we have an action ##S=S(a,b,c)## which is a functional of the fields ##a,\, b,\,## and ##c##. We denote the variation of ##S## wrt to a given field, say ##a##, i.e. ##\frac{\delta S}{\delta a}##, by ##E_a##. Then ##S## is gauge invariant when $$\delta S = \delta a E_a + \delta b E_b...
  4. joneall

    I Krauss's interpretation of gauge theory

    Lawrence Krauss, "The greatest story ever told ... so far", pp. 108-109. "Gauge symmetry in electromagnetism says that I can actually change my definition of what a positive charge is locally at each point of space without changing the fundamental laws associated with electric charge, as long...
  5. dextercioby

    A Wiesendanger's quantization of an SO(1,3) extension of GR

    Are you aware of the 3-article series of Wiesendanger's quantized extension of GR? This is open access: C Wiesendanger 2019 Class. Quantum Grav. 36 065015 and the two sequels linked to in the PDF. The question is if this work counts as a quantization of a reasonable extension or reformulation...
  6. J

    Model with SU(2) gauge symmetry and SO(3) global symmetry

    1.) The rule for the global ##SO(3)## transformation of the gauge vector field is ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## for ##\omega \in SO(3)##. The proof is by direct calculation. First, if ##A^i_{\mu} \to \omega_{ij}A^j_{\mu}## then ##F^i_{\mu \nu} \to \omega_{ij}F^j_{\mu\nu}##, so...
  7. JD_PM

    Modifying photon states to get a gauge symmetry

    I see that this procedure helps to get rid of the two extra degrees of freedom (due to the scalar and longitudinal photons) one firstly encounters while writing the electromagnetic field theory in a Lorentz-covariant way; it indeed shows that modifying the allowed admixtures of longitudinal and...
  8. QuasarBoy543298

    Exploring Gauge Symmetry in Classical Field Theory

    hi, I'm currently taking a classical field theory class (electromagnetism in the language of tensors and actions and etc) and we have just encountered the gauge symmetry, that is for the 4 vector potential we can add a gradient of some smooth function and get the same physics (if we take Aμ →...
  9. N

    Left invariant vector field under a gauge transformation

    Homework Statement For a left invariant vector field γ(t) = exp(tv). For a gauge transformation t -> t(xμ). Intuitively, what happens to the LIVF in the latter case? Is it just displaced to a different point in spacetime or something else? Homework EquationsThe Attempt at a Solution
  10. Q

    Yang Mills & Fiber Bundles Resource

    Hi everyone, Does anyone know of a good intuitive resource for learning Yang-Mills theory and Fiber Bundles? Ultimately my goal is to gain a geometric understanding of gauge theory generally. I have been studying differential forms and exterior calculus. Thanks!
  11. A

    I Understanding Gauge Symmetry: A Review of the Schrödinger Equation

    I have reviewed the various posts on gauge symmetry in particular this one which is now closed. In this post there is the following link:http://www.vttoth.com/CMS/physics-notes/124-the-principle-of-gauge-invariance. This is a good read. However, there is some clarification I need. The...
  12. J

    A Any good idea how non-abelian gauge symmetries emerge?

    I think the story where abelian, i.e. U(1), gauge symmetry comes from is pretty straight-forward: We describe massless spin 1 particles, which have only two physical degrees of freedom, with a spin 1 field, which is represented by a four-vector. This four-vector has 4 entries and therefore too...
  13. J

    A How can gauge symmetry be broken?

    Hello, I was reading about the Higgs mechanism and I must say that I did not really follow the argument of how the gauge symmetry is broken. I think that my problem has to do with the more general question of how does a gauge symmetry get broken in general? Thanks!
  14. F

    I Difference between global and local gauge symmetries

    The mantra in theoretical physics is that global gauge transformations are physical symmetries of a theory, whereas local gauge transformations are simply redundancies (representing redundant degrees of freedom (dof)) of a theory. My question is, what distinguishes them (other than being...
  15. J

    Quantum Book on gauge transformations/symmetry & geometrical phases?

    Hello! I will be attending a course on condensed matter physics with emphasis on geometrical phases and I was wondering if the are any good books on gauge transformations, gauge symmetry and geometrical phases that you know of. Thanks in advance!
  16. P

    Gauge symmetry for massless Klein-Gordon field

    Homework Statement I need to gauge the symmetry: \phi \rightarrow \phi + a(x) for the Lagrangian: L=\partial_\mu\phi\partial^\mu\phi Homework EquationsThe Attempt at a Solution We did this in class for the Dirac equation with a phase transformation and I understood the method, but...
  17. Andrea_G

    A Four Bosons vertex related to gauge symmetry

    Hi, is correct to say that there is no interaction between four photons because the gauge group of QED is U (1) while there are interactions of four gluons or four W's because the gauge group of QCD is SU (3) and EW's one is SU (2) xU (1)? I know that the interaction between four photons is not...
  18. T

    Why is gauge symmetry not a true symmetry?

    A symmetry of a physical system is a physical or mathematical feature of the system that is preserved or remains unchanged under some transformation. For example, the speed of light is an example of symmetry and its value will always will always remain the same no matter where and what...
  19. E

    Why must the Higgs' gauge symmetry be broken?

    The part I understand: I understand that the spontaneous symmetry breaking of the Higgs produces the 'Mexican hat' potential, with two non-zero stable equilibria. I understand that as the Higgs is a complex field, there exists a phase component of the field. Under gauge transformations of...
  20. Garlic

    Gauge Symmetry Breaking: Exploring Interactions at Low Temperatures

    Can there be interactions that are symmetric under low temperatures but exhibit spontaneous symmetry breaking under extremely low temperatures? (Maybe that symmetry breaking temperature is so low that it couldn't be discovered in experiments) Does electromagnetism split into electricity and...
  21. T

    Why does gauge fixing break gauge symmetry?

    By fixing a gauge (thus breaking orspending 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...
  22. Mentz114

    Understanding Gauge Symmetry in Massive Gravity: Analysis of Fierz-Pauli Action

    Refering to this paper "Theoretical Aspects of Massive Gravity" (http://arxiv.org/abs/1105.3735) about the spin-2 boson field and GR. The author uses the Fierz-Pauli action ( I quote the massless part) ##-\frac{1}{2}\partial_\lambda h_{\mu\nu}\partial^\lambda h^{\mu\nu} + \partial_\mu...
  23. K

    Gauge symmetry of cylindrical rod

    A twisted cylindrial rod has the cross sectional symmetry so that it's not posible to tell whether it is twisted or not without knowing if there is any torsional energy. now drawing a line on the surface of it can tell us whether or not it's twisted. It might not be a straight line.. there are...
  24. stevendaryl

    Is Choice of Spinor Representation a Gauge Symmetry?

    In the Dirac equation, the only thing about the gamma matrices that is "fixed" is the anticommutation rule: \gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 \eta^{\mu \nu} We can get an equivalent equation by taking a unitary matrix U and defining new spinors and gamma-matrices via...
  25. C

    Srednicki 58: EM current conservation & Gauge Symmetry

    Hi I am re-reading Srednicki's QFT. In chapter 58, he points out that the Noether current $$ j^\mu=e\bar{\Psi}\gamma^\mu\Psi$$ is only conserved when the fields are stationary, which is obvious from the derivation of the conservation law. Meanwhile he assumes that $$\partial _\mu...
  26. D

    U(1) Gauge Symmetry: What Informs Its Choice?

    So, I have a basic/general question here. I understand that, for example, the QED Langrangian has U(1) gauge symmetry. I also understand that this means (when you have written the Lagrangian with the covariant derivative) that if you transform the wavefunction (\psi \rightarrow e^{i \theta (x)}...
  27. Whovian

    Trouble explaining Gauge Symmetry

    I'm currently attempting to explain the concept of Gauge Symmetry to a friend. Copied and pasted pretty much directly from MathIM, (And the same applies for any other potential field, such as gravitational potential.) Would this be correct? I've tried explaining Gauge Symmetry multiple...
  28. T

    Gauge Symmetry in Quantum Mechanics (QM I) Explained

    Hi all, I'm taking graduate level QM I and trying to wrap my head around the notion of gauge symmetry. For some reason I've struggled with this concept more than others. I don't really have a specific question; I'm more looking to see if someone has a succinct explanation of the relevant...
  29. tom.stoer

    Photon helicity: Wigner's unitary rep. of Poincare group and gauge symmetry

    1) Since Wigner it is well known that for massless particles of spin s the physical states are labelled by helicity h = ±s; other states are absent. So e.g. for photons the physical states are labelled by |kμ, h> with kμkμ = 0 and h = ±1 and we have two d.o.f. 2) For gauge theories with...
  30. S

    Noether currents for local gauge symmetry

    hi everyone, I have been trying to understand gauge theory. I am familiar with the Noether's theorem applied in the context of simpler textbook cases like poincare invariant Lagrangians. This is my question: Are there Noether currents corresponding to the local gauge symmetries too and would...
  31. atyy

    Spontaneously broken gauge symmetry

    I have read 2 arguments that a gauge symmetry cannot be spontaneously broken. 1. Wen's textbook says a gauge symmetry is a by definition a "do nothing" transformation, so it cannot be broken. 2. Elitzur's theorem, eg.http://arxiv.org/abs/hep-ph/9810302v1 The first argument seems sound...
  32. C

    [Holography] Global symmetry in boundary corresponds to gauge symmetry in bulk?

    I hear the statement that global symmetries in the boundary field theory corresponds to gauge symmetries in the bulk. 1) Is this a generic statement that is expected to hold for all holography pairs? (Maldacena states this towards the end of his first lecture at PiTP2010, which was supposed to...
  33. L

    Gauge symmetry and renormalization

    Here and then I read gauge symmetry makes theories renormalizable. Unfortunately I could not find a satisfactory explanation why that so is. Could someone shed some light? thanks
  34. sweetser

    Gauge symmetry in EM by inspection

    Hello: I was under the impression that gauge symmetry was a property of the Lagrange density. Here is the Lagrangian for EM written out in its components: \begin{align*} \mathcal{L}_{EM} &= J\cdot A +\frac{1}{2}\left(B^2-E^2\right) \quad eq.~1\\ &=\rho \phi - Jx Ax - Jy Ay - Jz Az \\...
  35. N

    Why are there several gauge fixing choice for gauge symmetry fields?

    Please teach me this: For gauge symmetry fields,only one of any elementary subconfiguration of the whole configuration covers the all physics of the field.So we need to cut off the redundant configuration.It seem to me,in a loose sense,there is only one way to cut off the redundancy(the gauge...
  36. L

    Gauge symmetry for spin 1/2 fields

    Why is there no gauge symmetry for spin-1/2 fields? Has gauge symmetry to be related to spin-1 fields/ particles? thanks
  37. R

    Does Noether theorem apply to gauge symmetry?

    Basically, the title says it all. I've never heard of Noether charge corresponding to gauge symmetry of the Lagrangian. Is it because gauge symmetry isn't the "right type" of symmetry (one parameter continuous symmetry) so the Noether theorem doesn't apply to it?
  38. Q

    Deriving the needed wavefunction transformation for gauge symmetry?

    Homework Statement Take the Schrodinger equation for a point particle in a field: i\hbar \frac{\partial \Psi}{\partial t} = \frac{1}{2m}(-i\hbar\nabla - q\vec{A})^2\Psi + q\phi\Psi I'm supposed to determine what the transformation for Psi is that corresponds to the gauge transformation...
  39. K

    Exploring the Meaning of Gauge Symmetry

    I would like to hear an original explanation of gauge symmetry. What gauge symmetry really means and why it is needed to describe nature. I am more or less familiar with the standard treatment of electromagnetism and Yang Mills theories from QFT texts, but feel still unsatisfied since I have...
  40. arivero

    Kaluza Klein and gauge symmetry breaking.

    In standard, old-fashioned, Kaluza Klein theory we have new dimensionful parameters, the size of the compact dimensions, but they become dimensionless after quotient against the Plank size, so they become the adimensional coupling constants of the gauge groups associated to the symmetry of the...
  41. sweetser

    Exact Gauge Symmetry of the Standard Model

    Hello: The gauge symmetry of the standard model is written in authoritative places like wikipedia :-) as U(1)xSU(2)xSU(3). This would have 12 elements in its Lie algebra corresponding to one photon, W+, W- and W0 or Z, and the 8 gluons. I recall reading discussions that such a...
  42. A

    Understanding Local Gauge Symmetry

    Local Gauge Symmetry ?? Trying to understand local gauge symmetry ================================ I have an undergraduate degree in physics, so I know basic quantum mechanics, but that's all. Still, I'm trying to understannd the concept of local gauge symmetry. I would appreciate if...
  43. D

    Strings Chan-Paton U(N) gauge symmetry fractional winding number

    I understand why in the presence of a constant vector potential A=-\frac{\theta}{2 \pi R} along a compactified dimension (radius R) the canonical momentum of a -e charged particle changes to P=p-eA. Due to the single valuedness of the wavefunction [itex]\propto e^{iPX}[/tex] P should be...
  44. T

    Gauge symmetry and symmetry breaking

    How would one know in general, whether an original gauge symmetry in the theory is still gauge symmetrical after symmetry breaking? I mean is there a theorem or something like that? And the other way around, is there a general way of knowing whether there is the possibility of a hidden, i.e...
  45. A

    Definition for the term gauge symmetry

    Could somebody please give me a definition for the term gauge symmetry in contrast to any other symmetry? Is the decisive difference that a gauge symmetry is local i.e. a function of the coordinates in contrast to being constant? I would also appreciate it if it could be pointed out how the...
  46. K

    How does non-abelian gauge symmetry affect quark interactions?

    How does a non-abelian gauge symmetry lead to asymptotic freedom for quarks?