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--- advanced_tools:gauge_symmetry [2018/04/02 16:00]
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+++ advanced_tools:gauge_symmetry [2020/04/02 20:12] (current)
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@@ Line 1: / Line 1: @@
 ====== Gauge Symmetry ======
+//see also [[models:gauge_theory]] //
-//see also [[theories:gauge_theory]] //
 <tabbox Intuitive>
-A gauge symmetry is analogous to how we can describe something //within// one language through different words (synonyms). A description of the same thing in different languages is called a [[advanced_notions:quantum_field_theory:duality|Duality]].
+A gauge symmetry is analogous to how we can describe something //within// one language through different words (synonyms). A description of the same thing in different languages is called a [[advanced_notions:duality|Duality]].
@@ Line 12: / Line 10: @@
 When we describe things in physics, we have always some freedom in our description. For example, it doesn't matter what coordinate system we choose. It makes no difference where we choose the origin of the coordinate system or how it is oriented.
-The computations can be different in different coordinate systems and usually, one picks a coordinate system where the computation is especially simple. However, the physics that we are describing, of course, doesn't care about how we describe it. It stays the same, now matter how we choose our coordinate system.
+The computations can be different in different coordinate systems and usually, one picks a coordinate system where the computation is especially simple. However, the physics that we are describing, of course, doesn't care about how we describe it. It stays the same, no matter how we choose our coordinate system.
-In modern physics, we no longer describe what is happening merely through the position of objects at a given time, as we do it in classical mechanics. Instead, we use abstract objects called fields. The best theory of what is happening in nature at the most fundamental level is [[theories:quantum_field_theory]]. Like the electromagnetic field, these fields can get excited (think: we can produce a wave or ripple of the field). For example, when we excite the electron field we "produce" an electron.
+In modern physics, we no longer describe what is happening merely through the position of objects at a given time, as we do it in classical mechanics. Instead, we use abstract objects called fields. The best theory of what is happening in nature at the most fundamental level is [[theories:quantum_field_theory:canonical|quantum field theory]]. Like the electromagnetic field, these fields can get excited (think: we can produce a wave or ripple of the field). For example, when we excite the electron field we "produce" an electron.
 The fields themselves are abstract mathematical entities that are introduced as convenient mathematical tools. With these new mathematical entities comes a new kind of freedom. Completely analogous to how we have the freedom to choose the orientation and the location of the origin of our coordinate system, we now have freedom in how we define our fields.
@@ Line 46: / Line 44: @@
 Thus, by trading in a loop, we have gained 50 pounds.
-The loops we considered here are exactly analogous to the [[advanced_notions:quantum_field_theory:wilson_loops|Wilson loops]] used in [[theories:quantum_field_theory|quantum field theories]]. The //gauge freedom// corresponds here to the freedom to rescale the local currencies. For examples, England could introduce a new currency called "new-pound" and determine that 1 new-pound is worth 10 pounds. This wouldn't change the situation at the global money market at all because all banks would simply adjust their exchange rates:
+The loops we considered here are exactly analogous to the [[advanced_notions:quantum_field_theory:wilson_loops|Wilson loops]] used in [[theories:quantum_field_theory:canonical|quantum field theories]]. The //gauge freedom// corresponds here to the freedom to rescale the local currencies. For examples, England could introduce a new currency called "new-pound" and determine that 1 new-pound is worth 10 pounds. This wouldn't change the situation at the global money market at all because all banks would simply adjust their exchange rates:
   * 0.15 dollars = 1 new-pound,
@@ Line 74: / Line 72: @@
 When in the 19th century people tried to understand how electromagnetism works they also figured this out. However, they made also another intriguing discovery. When writing down the laws which govern electromagnetism, it turns out that electric and magnetic fields are intimately linked, and that they are just two sides of the same coin. That is the reason to call it electromagnetism.
-//In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as [[equations:maxwell_equations|Maxwell equations]]// in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with [[models:quantum_electrodynamics|quantum electro dynamics - QED]].
+//In the early 20th century it then became clear that both phenomena can be associated with a single particle, the photon. But then it was found that to characterize a photon only two numbers at each point in space and time are necessary. This implies that between the six numbers characterizing electric and magnetic fields relations exist. These are known as [[equations:maxwell_equations|Maxwell equations]]// in classical physics, or as quantum Maxwell dynamics in the quantum theory. If you would add, e. g., electrons to this theory, you would end up with [[models:standard_model:qed|quantum electro dynamics - QED]].
@@ Line 149: / Line 147: @@
 which is called a gauge transformation. This means immediately that the action is unchanged and that this transformation is a symmetry of the system.
+**Local Transformations**
+{{ :advanced_tools:localtransformations.png?nolink&600 |}}
 <tabbox Abstract>
 The group of gauge transformations $G$ means the bundles automorphisms which preserve the Lagrangian. ([[http://www.mathunion.org/ICM/ICM1978.2/Main/icm1978.2.0881.0886.ocr.pdf|Source]])
@@ Line 182: / Line 183: @@
 \end{align}
-This comes about when one considers [[equations:gauss_law|Gauss law]] to identify physical states. Such physical states are invariant under $\mathcal{G}_\star$ and thus this subgroup connects physically redundant variables in the theory.
+This comes about when one considers [[formulas:gauss_law|Gauss law]] to identify physical states. Such physical states are invariant under $\mathcal{G}_\star$ and thus this subgroup connects physically redundant variables in the theory.
 Since the elements of $\mathcal G$ go only to a constant, which is not necessarily $1$ at spatial infinity, we have
@@ Line 225: / Line 226: @@
 <tabbox Why is it interesting?>
-Gauge symmetries are at the heart of the best theory of fundamental interactions, the [[models:standard_model|standard model]] of particle physics. Theories that make use of gauge symmetry are commonly called [[theories:gauge_theory|gauge theories]].
+Gauge symmetries are at the heart of the best theory of fundamental interactions, the [[models:standard_model|standard model]] of particle physics. Theories that make use of gauge symmetry are commonly called [[models:gauge_theory|gauge theories]].
 In addition to this application, gauge symmetry can also be useful to understand finance. This is shown, for example, in
@@ Line 261: / Line 262: @@
 <tabbox FAQ>
---> Is a gauge symmetry really a symmetry?#
-No!
-<blockquote>gauge symmetries aren’t real symmetries: they are merely
-redundancies in our description of the system.<cite>[[http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf|David Tong]]</cite></blockquote>
-<blockquote>
-[Gauge symmetry], thinking about it as a symmetry is a bad idea, thinking about it as being broken is a bad idea.
-<cite>[[https://youtu.be/XM4rsPnlZyg?t=23m46s|Duality and emergent gauge symmetry - Nathan Seiberg]]</cite>
-</blockquote>
-<blockquote>
-The problem with gauge symmetry is that it is not a symmetry in the sense of quantum mechanics. A symmetry is the invariance of the Hamiltonian under transformations of quantum states, which are elements of a Hilbert space. Gauge symmetry is not a symmetry because the corresponding transformation does not change the quantum states. Gauge symmetry acts trivially on the Hilbert space and does not relate physically distinct states. A gauge transformation is like a book by James Joyce: it seems that something is going on, but nothing really happens.
-Gauge symmetry is the statement that certain degrees of freedom do not exist in the theory. This is why gauge symmetry corresponds only to as a redundancy of the theory description. The non-symmetry nature of gauge symmetry explains why gauge symmetry, unlike global symmetry, cannot be broken by adding local operators to the action: gauge symmetry is exact at all scales. The only way to “break” gauge symmetry is adding to the theory the missing degrees of freedom, but this operation is not simply a deformation of the theory (as the case of adding local operators to an action with global symmetry) but corresponds to considering an altogether different theory. The non-symmetry nature of gauge symmetry also explains trivially the physical content of the Higgs theorem. For a spontaneously-broken global symmetry, an infinite number of vacuum states are related by the symmetry transformation. This leads to the massless modes dictated by the Goldstone theorem. In a spontaneously-broken gauge symmetry, there is a single physical vacuum and thus there are no massless Goldstones. Gauge symmetry does not provide an exception to the Goldstone theorem, simply because there is no symmetry to start with.
-For gauge symmetry, the word ‘symmetry’ is a misnomer, much as ‘broken’ is a misnomer for spontaneously broken symmetry. But as long as the physical meaning is clear, any terminology is acceptable in human language. The important aspect is that the mathematical language of gauge symmetry (both in the linear and non-linear versions) is extremely pow- erful in physics and permeates the Standard Model, general relativity, and many systems in condensed matter. As the redundancy of degrees of freedom is mathematically described by the same group theory used for quantum symmetries, the use of the word ‘symmetry’ seems particularly forgivable.
-Does this necessarily make gauge symmetry a fundamental element in the UV? The property of gauge symmetry of being – by construction – valid at all energy scales may naively suggest that gauge symmetry must be an ingredient of any UV theory from which the Standard Model and general relativity are derived. On the contrary, many examples have been constructed – from duality to condensed-matter systems – where gauge symmetry is not fundamental, but only an [[advanced_notions:emergence|emergent property]] of the effective theory [41]. Gauge symmetry could emerge in the IR, without being present in the UV theory. If this is the case, gauge symmetry is not the key that will unlock the mysteries of nature at the most fundamental level. The concept of symmetry has given much to particle physics, but it could be that it is running out of fuel and that, in the post-naturalness era, new concepts will replace symmetry as guiding principles.
-<cite>https://arxiv.org/pdf/1710.07663.pdf#page23</cite>
-</blockquote>
-See https://youtu.be/XM4rsPnlZyg?t=18m38s
-<blockquote>
-But there are several reasons not to accept this view. First of all terminology. When we say gauge symmetry, this is really a misnomer. It's a misnomer because in physics gauge symmetry is not a symmetry. It is not a symmetry of anything. Symmetry is a set of transformations that act on physical observables. They act on the Hilbert space. The Hilbert space is always gauge invariant. So the **gauge symmetry doesn't even act on the Hilbert space**. So it's not a symmetry of anything. [...] Second, **gauge symmetry can be made to look trivial**. So, I'll give one trivial example and then I'll make it more elaborate... [explains the Stückelberg mechanism, where one introduces a Stückelberg field to make a non U(1) gauge invariant Lagrangian, gauge invariant] This is almost like a fake... This gauge symmetry is what we would call emergent, except that in this case it is completely trivial. The second thing which is wrong about gauge symmetry, which suggests that it's not fundamental is that, it started in condensed matter physics, people talked about spontaneous symmetry breaking. That was crucial in the context of superconductivity and superfluidity and so forth. And the recent Nobel price in physics was also associated with spontaneous gauge symmetry breaking. That of Higgs, and Englert. **This is all very nice and physicists love to talk about spontaneous symmetry breaking, but this is a bit too naive**. First of all I've already emphasized that a gauge symmetry is not a symmetry. And since it is not a symmetry, how could it possibly be broken. You can break a symmetry that exists, but **you cannot break a symmetry that does not exist**. [[https://physics.stackexchange.com/questions/321857/why-do-we-assume-the-spatial-volume-is-infinite|Second, the phenomenon of spontaneous symmetry breaking is often associated with the fact that the system goes to infinity. Concretely in quantum mechanics, you never have symmetry breaking.]] It is only in quantum field theory or statistical mechanics, where we have volume going to infinity we have an infinite number of degrees of freedom and there we have this phenomenon of spontaneous symmetry breaking. That's not true for gauge theories. For gauge theories, we have a lot of symmetry. At every point of space we have a separate symmetry. But the number of degrees of freedom that transform under a given symmetry transformation is always finite. Nothing goes off to infinity. **So the gauge symmetry cannot be spontaneously broken.** The ground state is always unique. Or if you wish, all these would-be separate ground states are all related to each other by a gauge transformation. [...] I said that gauge symmetry cannot be ultimate symmetry because it's so big, there is a separate transformation at every point in space. So the breaking of a gauge theory cannot happen, I can use a phrase from the financial crisis in 2008 that **a gauge symmetry is so big, it's too big to fail**.
-<cite>Duality and emergent gauge symmetry - Nathan Seiberg</cite>
-</blockquote>
-See also Seiberg's slides starting at page 30 here http://research.ipmu.jp/seminar/sysimg/seminar/1607.pdf:
-<blockquote>
-Gauge symmetry is deep
-•Largest symmetry (a group for each point in spacetime)
-•Useful in making the theory manifestly Lorentz invariant, unitary and local (and hence causal)
-But
-•Because of Gauss law the Hilbert space is gauge invariant.( More precisely, it is invariant under small gauge transformation; large gauge transformations are central.)
-•Hence:gauge symmetry is not asymmetry.
-• It does not act on anything.
-• A better phrase is gauge redundancy.
-Gauge symmetries cannot break
-•Not a symmetry and hence cannot break
-•For spontaneous symmetry breaking we need an infinite number of degrees of freedom transforming under the symmetry.  Not here.
-•This is the deep reason there is no massless Nambu-Goldstone boson when gauge symmetries are “broken.”
-Gauge symmetries cannot break For weakly coupled systems (e.g. Landau-Ginsburg theory of  superconductivity, or the weak interactions) the language of spontaneous gauge symmetry breaking is appropriate and extremely useful[Stueckelberg,Anderson,Brout, Englert,Higgs].
-Global symmetries can emerge as accidental
-symmetries at long distance.
-Then they are approximate.
-**Exact gauge
-symmetries can be emergent.**
-Examples of emergent gauge symmetry...
-<cite>http://research.ipmu.jp/seminar/sysimg/seminar/1607.pdf</cite>
-</blockquote>
-<blockquote>
-Gauge symmetries are
-properly to be thought of as not being symmetries at all, but rather redundancies in
-our description of the system [[https://projecteuclid.org/euclid.cmp/1103920387|1]]. The true configuration space of a (3 + 1)-
-dimensional gauge theory is the quotient $\mathcal{A}^3/\mathcal{G}^3$ of gauge potentials in $A_0=0$ gauge modulo three-dimensional gauge transformations. When gauge degrees of
-freedom become [[advanced_notions:quantum_field_theory:anomalies|anomalous]], we find that they are not redundant after all.
-<cite>[[https://projecteuclid.org/download/pdf_1/euclid.cmp/1103942612|Hamiltonian Interpretation of Anomalies]] by Philip Nelson and Luis Alvarez-Gaume</cite>
-</blockquote>
-<blockquote>
-From the modern point of view, then, **gauge symmetry is merely a useful redundancy** for describing the physics of interacting massless particle of spin 1 or 2, **tied to the specific formalism of Feynman diagrams**, that makes locality and unitarity as manifest as possible.
-<cite>https://arxiv.org/pdf/1612.02797.pdf</cite>
-</blockquote>
-<blockquote>
-Gauge invariance is not physical. It is not observable and is not a symmetry of nature.
-Global symmetries are physical, since they have physical consequences, namely conservation of charge. That is, we measure the total charge in a region, and if nothing leaves that
-region, whenever we measure it again the total charge will be exactly the same. There is no
-such thing that you can actually measure associated with gauge invariance. We introduce
-gauge invariance to have a local description of massless spin-1 particles. The existence of these particles, with only two polarizations, is physical, but the gauge invariance is merely
-a redundancy of description we introduce to be able to describe the theory with a local
-Lagrangian. A few examples may help drive this point home. First of all, an easy way to see that
-gauge invariance is not physical is that we can choose any gauge, and the physics is going to
-be exactly the same. In fact, we have to choose a gauge to do any computations. Therefore,
-there cannot be any physics associated with this artificial symmetry.
-<cite>Quantum Field Theory and the Standard Model by Matthew Schwartz</cite>
-</blockquote>
-<--
@@ Line 586: / Line 479: @@
---> What makes Yang-Mills theories so difficult to work with?#
+--> What conserved quantities follow from gauge symmetries?#
-<blockquote>The hardest problem in Yang-Mills theory is the problem of reduction of the gauge symmetry (redundancy); i.e. the characterization of the orbit space of gauge potentials modulo gauge transformations.
-In 3+1 dimensions, this space is awfully complicated both geometrically and topologically. The solution of this problem should shed light to the longstanding open problems in Yang-Mills theory such as the mass gap and confinement. The reduced orbit space is infinite dimensional and not even a manifold.
+Using [[theorems:noethers_theorems|Noether's first theorem]] we find that the conserved charges that would follow from invariance under gauge transformations are identically zero.  This is shown, for example, in section 3.4.1 [[http://math.ucr.edu/home/baez/classical/texfiles/2005/book/classical.pdf|here]].
-If we knew a solution of this problem, we could, in principle, quantize the reduced configuration space (by means of geometric quantization, which is itself a formidable problem) and obtain the quantum Yang-Mills which should reflect the above conjectured properties.
+However, using Noether's second theorem we can derive relations between our equations of motion, that are known as Bianchi identities.
-Most of the known methods (except lattice regularization) use gauge fixing (together with Feddeev-Popov ghosts and BRST) to reduce the gauge redundancy. This method reduces the gauge redundancy only approximately as it suffers from the problem of Gribov copies. It is believed that this approximation is valid only in perturbation theory. There are several non-perturbative approximations such as the Faddeev-Niemi theory, but their connection to Yang-Mills is only heuristic.
-Two physicists: Gurd Rudolf and Matthias Schmidt (together with a number of collaborators) are working on several methods to tackle this hard problem by reducing the gauge redundancy without gauge fixing. They have many publications on the subject. They use mainly methods of geometry and topology. Their efforts led to certain classification results of the yang-Mills gauge orbit. They wrote a book named Differential geometry and Mathematical physics (Part 1 , Part 2).
+<--
-In the book, they give a detailed account of the basics of geometry and topology relevant to the Yang-Mills theory in a rigorous mathematical presentation. The entire book can be viewed, however, as an introduction to the last two chapters of Part 2 where they give account of some of their results in the classification and quantization of the Yang-Mills theory. This subject is very hard. Their research is still ongoing. They have results for certain toy models, such as a lattice of a single plaquette. They tackle both problems of the classical characterization of the orbit space and its quantization for these models. The book covers many advanced topics and can be a useful reference for a physicist interested in Yang-Mills theory research, and quantization.<cite>https://physics.stackexchange.com/a/368458/37286</cite></blockquote>
+--> Is a gauge symmetry really a symmetry?#
+No!
+<blockquote>gauge symmetries aren’t real symmetries: they are merely
+redundancies in our description of the system.<cite>[[http://www.damtp.cam.ac.uk/user/tong/qhe/qhe.pdf|David Tong]]</cite></blockquote>
+<blockquote>
+[Gauge symmetry], thinking about it as a symmetry is a bad idea, thinking about it as being broken is a bad idea.
+<cite>[[https://youtu.be/XM4rsPnlZyg?t=23m46s|Duality and emergent gauge symmetry - Nathan Seiberg]]</cite>
+</blockquote>
+<blockquote>
+The problem with gauge symmetry is that it is not a symmetry in the sense of quantum mechanics. A symmetry is the invariance of the Hamiltonian under transformations of quantum states, which are elements of a Hilbert space. Gauge symmetry is not a symmetry because the corresponding transformation does not change the quantum states. Gauge symmetry acts trivially on the Hilbert space and does not relate physically distinct states. A gauge transformation is like a book by James Joyce: it seems that something is going on, but nothing really happens.
+Gauge symmetry is the statement that certain degrees of freedom do not exist in the theory. This is why gauge symmetry corresponds only to as a redundancy of the theory description. The non-symmetry nature of gauge symmetry explains why gauge symmetry, unlike global symmetry, cannot be broken by adding local operators to the action: gauge symmetry is exact at all scales. The only way to “break” gauge symmetry is adding to the theory the missing degrees of freedom, but this operation is not simply a deformation of the theory (as the case of adding local operators to an action with global symmetry) but corresponds to considering an altogether different theory. The non-symmetry nature of gauge symmetry also explains trivially the physical content of the Higgs theorem. For a spontaneously-broken global symmetry, an infinite number of vacuum states are related by the symmetry transformation. This leads to the massless modes dictated by the Goldstone theorem. In a spontaneously-broken gauge symmetry, there is a single physical vacuum and thus there are no massless Goldstones. Gauge symmetry does not provide an exception to the Goldstone theorem, simply because there is no symmetry to start with.
+For gauge symmetry, the word ‘symmetry’ is a misnomer, much as ‘broken’ is a misnomer for spontaneously broken symmetry. But as long as the physical meaning is clear, any terminology is acceptable in human language. The important aspect is that the mathematical language of gauge symmetry (both in the linear and non-linear versions) is extremely pow- erful in physics and permeates the Standard Model, general relativity, and many systems in condensed matter. As the redundancy of degrees of freedom is mathematically described by the same group theory used for quantum symmetries, the use of the word ‘symmetry’ seems particularly forgivable.
+Does this necessarily make gauge symmetry a fundamental element in the UV? The property of gauge symmetry of being – by construction – valid at all energy scales may naively suggest that gauge symmetry must be an ingredient of any UV theory from which the Standard Model and general relativity are derived. On the contrary, many examples have been constructed – from duality to condensed-matter systems – where gauge symmetry is not fundamental, but only an [[advanced_notions:emergence|emergent property]] of the effective theory [41]. Gauge symmetry could emerge in the IR, without being present in the UV theory. If this is the case, gauge symmetry is not the key that will unlock the mysteries of nature at the most fundamental level. The concept of symmetry has given much to particle physics, but it could be that it is running out of fuel and that, in the post-naturalness era, new concepts will replace symmetry as guiding principles.
+<cite>https://arxiv.org/pdf/1710.07663.pdf#page23</cite>
+</blockquote>
+See https://youtu.be/XM4rsPnlZyg?t=18m38s
+<blockquote>
+But there are several reasons not to accept this view. First of all terminology. When we say gauge symmetry, this is really a misnomer. It's a misnomer because in physics gauge symmetry is not a symmetry. It is not a symmetry of anything. Symmetry is a set of transformations that act on physical observables. They act on the Hilbert space. The Hilbert space is always gauge invariant. So the **gauge symmetry doesn't even act on the Hilbert space**. So it's not a symmetry of anything. [...] Second, **gauge symmetry can be made to look trivial**. So, I'll give one trivial example and then I'll make it more elaborate... [explains the Stückelberg mechanism, where one introduces a Stückelberg field to make a non U(1) gauge invariant Lagrangian, gauge invariant] This is almost like a fake... This gauge symmetry is what we would call emergent, except that in this case it is completely trivial. The second thing which is wrong about gauge symmetry, which suggests that it's not fundamental is that, it started in condensed matter physics, people talked about spontaneous symmetry breaking. That was crucial in the context of superconductivity and superfluidity and so forth. And the recent Nobel price in physics was also associated with spontaneous gauge symmetry breaking. That of Higgs, and Englert. **This is all very nice and physicists love to talk about spontaneous symmetry breaking, but this is a bit too naive**. First of all I've already emphasized that a gauge symmetry is not a symmetry. And since it is not a symmetry, how could it possibly be broken. You can break a symmetry that exists, but **you cannot break a symmetry that does not exist**. [[https://physics.stackexchange.com/questions/321857/why-do-we-assume-the-spatial-volume-is-infinite|Second, the phenomenon of spontaneous symmetry breaking is often associated with the fact that the system goes to infinity. Concretely in quantum mechanics, you never have symmetry breaking.]] It is only in quantum field theory or statistical mechanics, where we have volume going to infinity we have an infinite number of degrees of freedom and there we have this phenomenon of spontaneous symmetry breaking. That's not true for gauge theories. For gauge theories, we have a lot of symmetry. At every point of space we have a separate symmetry. But the number of degrees of freedom that transform under a given symmetry transformation is always finite. Nothing goes off to infinity. **So the gauge symmetry cannot be spontaneously broken.** The ground state is always unique. Or if you wish, all these would-be separate ground states are all related to each other by a gauge transformation. [...] I said that gauge symmetry cannot be ultimate symmetry because it's so big, there is a separate transformation at every point in space. So the breaking of a gauge theory cannot happen, I can use a phrase from the financial crisis in 2008 that **a gauge symmetry is so big, it's too big to fail**.
+<cite>Duality and emergent gauge symmetry - Nathan Seiberg</cite>
+</blockquote>
+See also Seiberg's slides starting at page 30 here http://research.ipmu.jp/seminar/sysimg/seminar/1607.pdf:
-<--
+<blockquote>
+Gauge symmetry is deep
+•Largest symmetry (a group for each point in spacetime)
+•Useful in making the theory manifestly Lorentz invariant, unitary and local (and hence causal)
+But
+•Because of Gauss law the Hilbert space is gauge invariant.( More precisely, it is invariant under small gauge transformation; large gauge transformations are central.)
+•Hence:gauge symmetry is not asymmetry.
+• It does not act on anything.
+• A better phrase is gauge redundancy.
+Gauge symmetries cannot break
+•Not a symmetry and hence cannot break
+•For spontaneous symmetry breaking we need an infinite number of degrees of freedom transforming under the symmetry.  Not here.
+•This is the deep reason there is no massless Nambu-Goldstone boson when gauge symmetries are “broken.”
+Gauge symmetries cannot break For weakly coupled systems (e.g. Landau-Ginsburg theory of  superconductivity, or the weak interactions) the language of spontaneous gauge symmetry breaking is appropriate and extremely useful[Stueckelberg,Anderson,Brout, Englert,Higgs].
+Global symmetries can emerge as accidental
+symmetries at long distance.
+Then they are approximate.
+**Exact gauge
+symmetries can be emergent.**
+Examples of emergent gauge symmetry...
+<cite>http://research.ipmu.jp/seminar/sysimg/seminar/1607.pdf</cite>
+</blockquote>
+<blockquote>
+Gauge symmetries are
+properly to be thought of as not being symmetries at all, but rather redundancies in
+our description of the system [[https://projecteuclid.org/euclid.cmp/1103920387|1]]. The true configuration space of a (3 + 1)-
+dimensional gauge theory is the quotient $\mathcal{A}^3/\mathcal{G}^3$ of gauge potentials in $A_0=0$ gauge modulo three-dimensional gauge transformations. When gauge degrees of
+freedom become [[advanced_notions:quantum_field_theory:anomalies|anomalous]], we find that they are not redundant after all.
+<cite>[[https://projecteuclid.org/download/pdf_1/euclid.cmp/1103942612|Hamiltonian Interpretation of Anomalies]] by Philip Nelson and Luis Alvarez-Gaume</cite>
+</blockquote>
+<blockquote>
+From the modern point of view, then, **gauge symmetry is merely a useful redundancy** for describing the physics of interacting massless particle of spin 1 or 2, **tied to the specific formalism of Feynman diagrams**, that makes locality and unitarity as manifest as possible.
+<cite>https://arxiv.org/pdf/1612.02797.pdf</cite>
+</blockquote>
+<blockquote>
+Gauge invariance is not physical. It is not observable and is not a symmetry of nature.
+Global symmetries are physical, since they have physical consequences, namely conservation of charge. That is, we measure the total charge in a region, and if nothing leaves that
+region, whenever we measure it again the total charge will be exactly the same. There is no
+such thing that you can actually measure associated with gauge invariance. We introduce
+gauge invariance to have a local description of massless spin-1 particles. The existence of these particles, with only two polarizations, is physical, but the gauge invariance is merely
+a redundancy of description we introduce to be able to describe the theory with a local
+Lagrangian. A few examples may help drive this point home. First of all, an easy way to see that
+gauge invariance is not physical is that we can choose any gauge, and the physics is going to
+be exactly the same. In fact, we have to choose a gauge to do any computations. Therefore,
+there cannot be any physics associated with this artificial symmetry.
+<cite>Quantum Field Theory and the Standard Model by Matthew Schwartz</cite>
+</blockquote>
+<--
 <tabbox History>

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