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longitudinal gauge degree of freedom
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How many degrees of freedom does the photon have in 2+1 dimensions ?
In ordinary theory of QED, the photon has two degrees of freedom, so when we want to quantize the electromagnetic field we impose two conditions to eliminate two degrees of freedom and get a photon with two degrees of freedom.
Do we impose two conditions to quantize the electromagnetic field in 2+1 dimensions?
Do we impose two conditions to quantize the electromagnetic field in 2+1 dimensions?
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Oct 11, 2012 · Modified Oct 11, 2012 by the commenter.
Jakson M. Fonseca · Universidade Federal de Viçosa (UFV)In 2+1 dimensions like in 3+1, we have 2 conditions that fixes the number of degrees of freedon: gauge invariance and motion equation, then only one of the three components of the potencial is physical and the photon has only one degree freedon in 2+1 dimensions.
Brian P Dolan · National University of Ireland, MaynoothThe answer depends on what 2+1 dimensional theory you wish to consider. Jakson's answer above is perfectly correct for the 2+1 dimensional version of Maxwell's theory, with action E^2 - B^2. But in 2+1 dimensions the dynamics of the photon can be changed in a way that is not possible in 3+1 dimensions, by adding a Chern-Simons term to the action. When this is done a photon in 2+1 dimensions can acquire a mass
in a consistent way and it would then have two degrees of freedom, one longitudinal and one transverse. This is not possible in 3+1 dimensions. Without the Chern-Simons term a 2+1 dimensional photon is massless, it only has one transverse degree of freedom as Jakson said.- Jakson M. Fonseca · Universidade Federal de Viçosa (UFV)I think this is a very confusion discussion. In a topological Maxwell-Chern-Simons
theory we have two conditions that fixes the number of degree freedon: gauge invariance and motion equation, like in massless case, but in a pure Maxwell case the spin of the photon is zero, being the photon in 2+1 dimensions a scalar particle. When we considerer the Maxwell-Chern-Simons action the photon has two possiblities that dependes on the mass sinal, and the photon's spin can be +1 (positive mass) or -1 (negative mass), but only one is possible, being the Maxwell-Chern-Simons one particle with only one degree of freedon. I think that the article "Topologically Massive Gauge Theories" (Annals of Physics 281, 409 449 (2000)) is a good reference about this question. 
Tomer Shacham · Hebrew University of JerusalemAs Brian said, this question is theory dependent. I would like to note that another possibility exists, that of only a CS term with no Maxwell term. In this case, the photon has no physical degree of freedom at all. With a Maxwell term, only one physical degree of freedom exists. (A theory with just the Maxwell term without the CS term doesn't really make sense.)-
James T. Wheeler · Utah State UniversityThe photon is not theory dependent, it is the quantization of the Maxwell field. The more general category to which the other comments apply is 'vector field'. For a general vector field in n-dimensions, there will be n degrees of freedom in general. Examples include the Proca lagrangian (Maxwell plus a mass term), and the topological case above. One of these is longitudinal and corresponds to mass. A massless vector field will have n-1 degrees of freedom. The photon, however, is a U(1) gauge particle and gauge theory imposes strong constraints on the form of the action, which must be invariant under the gauge transformation. A mass term ( m^2 A^a ) is inconsistent with the gauge transformation so gauge particles are always massless (this is why the Higgs is required in the standard model). The gauge dependence allows us to replace A with A+df, and this allows us to remove two degrees of freedom. In 2+1 dimensions, this leaves 1 degree of freedom.
Cracking the Quantum Code of the Universe
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John Moffat - 2014 - Science
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The “Higgs field” (‘scalar field’) is a postulate that is nothing more than a
theoretical variation upon the Aether concept. As many know conceptions of the
Aether likewise posits a ‘universal field’ or ‘universal essence’ by way of
which particles acquire their substance and sustenance. So, how does one explain
the origin of “mass”? One simply invents a ‘universal field’ consisting of
quantum “particles” (or “ripples”) of much greater “mass” which smaller
particles can supposedly ‘couple to’ as they interact with said “field”. The
electromagnetic field has its quantum particle i.e. the “photon” so why not
invent a likewise “field” for “mass”.
How is trying to explain the acquisition of “mass” as being the result of interactions with a hypothetical “field” of particles having greater “mass” supposed to be an explanation for how particles acquire their “mass”? That doesn’t seem explain the supposed ‘origin’ and/or acquisition of “mass”’. Yet, the Aether (“charge”) functions along similar lines in some respects but it is not by way of a ‘coupling’ dynamic.
L. Krauss goes on to assert that Life and Existence is all just a happy accident - “We’re only here because we have mass; and it’s because of the Higgs field”. This is like listening to an astrophysicist who asserts that ‘We’re only here because we are stardust’. Or, ‘We’re only here because there are neutrinos’ – or something. Then:
Those who espouse the existence of Aether have proposed this fundamental for centuries and little known, or understood as such, a variety of the most ancient mythologies across the globe utilize metaphor and allegory in relation to the activities of said Fundamental Substance predating even the origin of the conceptual term ‘Aether’ via Aristotle. Where hast thou been? And then:
… “Invent” something? The effort only reinvents the age’s old concept of the Aether whether termed as such; or not, similar in fashion to Einstein’s reinvention of same as a “space time fabric”. Ten or twenty billion dollars is being spent to postulate by way of inference the same basic fundamental principle?? Seriously? And then; the shameless plug from B. Green (I actually like B.Green):
Well of course, as long as funding is continued so that physicist can have a feel good moment about even the inference of a ‘universal field’ even though the same idea had been put forth long ago. Next up, Supersymmetry, which doubles the number of “particles” in the particle zoo again with “superpartners”, waiting in the wings to justify the continued funding and the further pursuit of more “mass”-ively expensive and fantastically short-lived ‘inference findings’. Can anyone spare a few moments to solve the supposed “energy crisis” with any of these inferred 'findings'?
So, consider this brief comparison:
Key concepts underlined: Yet, for someone to put forth the Aether and posit that the One Primary Substance undergoes Motion (‘fluctuations” above) to such extent that from ‘self-induced’-‘self-differentiations’ (“self-energy” above) due to these Motions, “matter” (“real electrons” above as an example) may be ‘precipitated’ or ‘materialize’ (just as described above) - destined to return to That Source (“annihilating back to the vacuum” above) that the cycle continues - becomes somehow unusual?? How so when this is principally and conceptually what has been put forth in other terms?
Consider it again:
By the power of Philosophy, Experiment, and Reason the existence of a Fundamental Essence was derived long before latter-day “modern physics”. The principle of a fundamental “universal field”, as a principle, is not new and far greater work has been done with much less and NO public dollars (Nikola Tesla, Aetherometry & Eric Dollard etc). Goodness me Tesla had nothing but brass, copper, and wood but far greater noble aims having come to the same conclusion.
I don't mean to sound cynical but am I missing something (?) - because I’m seriously not impressed with any of this. Someone help me out here.
If there be nothing new, but that which is
Hath been before, how are our brains beguiled,
Which, labouring for invention, bear amiss
The second burden of a former child. - Shakespeare
How is trying to explain the acquisition of “mass” as being the result of interactions with a hypothetical “field” of particles having greater “mass” supposed to be an explanation for how particles acquire their “mass”? That doesn’t seem explain the supposed ‘origin’ and/or acquisition of “mass”’. Yet, the Aether (“charge”) functions along similar lines in some respects but it is not by way of a ‘coupling’ dynamic.
L. Krauss goes on to assert that Life and Existence is all just a happy accident - “We’re only here because we have mass; and it’s because of the Higgs field”. This is like listening to an astrophysicist who asserts that ‘We’re only here because we are stardust’. Or, ‘We’re only here because there are neutrinos’ – or something. Then:
I’m amazed if it’s there because it’s really a theoretical edifice that’s been built over fifty years that requires something fantastical: this field throughout space and it seems almost too good to be true.
Those who espouse the existence of Aether have proposed this fundamental for centuries and little known, or understood as such, a variety of the most ancient mythologies across the globe utilize metaphor and allegory in relation to the activities of said Fundamental Substance predating even the origin of the conceptual term ‘Aether’ via Aristotle. Where hast thou been? And then:
When you’re a theoretical physicist like the two of us its weird when you’re sitting around at night and you invent something; to think that maybe Nature accepts that, or follows that rule.
… “Invent” something? The effort only reinvents the age’s old concept of the Aether whether termed as such; or not, similar in fashion to Einstein’s reinvention of same as a “space time fabric”. Ten or twenty billion dollars is being spent to postulate by way of inference the same basic fundamental principle?? Seriously? And then; the shameless plug from B. Green (I actually like B.Green):
Certainly from the perspective of convincing funding agencies to continue this line of research it’s better to have something that you can point to than something that is a null result; we’re excited about that.
Well of course, as long as funding is continued so that physicist can have a feel good moment about even the inference of a ‘universal field’ even though the same idea had been put forth long ago. Next up, Supersymmetry, which doubles the number of “particles” in the particle zoo again with “superpartners”, waiting in the wings to justify the continued funding and the further pursuit of more “mass”-ively expensive and fantastically short-lived ‘inference findings’. Can anyone spare a few moments to solve the supposed “energy crisis” with any of these inferred 'findings'?
So, consider this brief comparison:
A charged particle inevitably produces an electric potential around it, and it also feels the potential created by itself. This leads to an infinite "self-energy" of the electron.
(...)
Once there exists anti-matter, which can annihilate matter or be created with matter, what we consider to be an empty vacuum undergoes a fluctuation to produce a pair of electron and positron together with photon, annihilating back to vacuum within the time interval allowed by the uncertainty principle (a). In addition to the effect of the electric potential on itself (b), the electron can annihilate with a positron in the fluctuation, leaving the electon originally in the fluctuation to materialize as a real electron (c). It turns out, these two contributions to the energy of the electron almost nearly cancel with each other. The small size of the electron was made consistent with electromagnetism thanks to quantum mechanics and the existence of anti-matter. – Introduction to Supersymmetry
Key concepts underlined: Yet, for someone to put forth the Aether and posit that the One Primary Substance undergoes Motion (‘fluctuations” above) to such extent that from ‘self-induced’-‘self-differentiations’ (“self-energy” above) due to these Motions, “matter” (“real electrons” above as an example) may be ‘precipitated’ or ‘materialize’ (just as described above) - destined to return to That Source (“annihilating back to the vacuum” above) that the cycle continues - becomes somehow unusual?? How so when this is principally and conceptually what has been put forth in other terms?
Consider it again:
The primary substance, thrown into infinitesimal whirls of prodigious velocity, becomes gross matter; the force subsiding, the motion ceases and matter disappears, reverting to the primary substance." - Nikola Tesla
By the power of Philosophy, Experiment, and Reason the existence of a Fundamental Essence was derived long before latter-day “modern physics”. The principle of a fundamental “universal field”, as a principle, is not new and far greater work has been done with much less and NO public dollars (Nikola Tesla, Aetherometry & Eric Dollard etc). Goodness me Tesla had nothing but brass, copper, and wood but far greater noble aims having come to the same conclusion.
I don't mean to sound cynical but am I missing something (?) - because I’m seriously not impressed with any of this. Someone help me out here.
If there be nothing new, but that which is
Hath been before, how are our brains beguiled,
Which, labouring for invention, bear amiss
The second burden of a former child. - Shakespeare
All Answers (6)