A gauge transformation rotates the phase of the condensate by an amount which changes from point to point, and shifts the vector potential by a gradient:. But when there is a condensate, the phase of the condensate defines a preferred choice of phase. Fixing the choice of gauge so that the condensate has the same phase everywhere, the electromagnetic field energy has an extra term,. When this term is present, electromagnetic interactions become short-ranged. Every field mode, no matter how long the wavelength, oscillates with a nonzero frequency. The lowest frequency can be read off from the energy of a long wavelength A mode,.
In an actual superconductor, the charged particles are electrons, which are fermions not bosons. So in order to have superconductivity, the electrons need to somehow bind into Cooper pairs. The pairing in a normal superconductor is due to lattice vibrations, and is in fact very weak; this means that the pairs are very loosely bound. Gauge invariance means that certain transformations of the gauge field do not change the energy at all.
If an arbitrary gradient is added to A , the energy of the field is exactly the same. This makes it difficult to add a mass term, because a mass term tends to push the field toward the value zero. But the zero value of the vector potential is not a gauge invariant idea.
What is zero in one gauge is nonzero in another. So in order to give mass to a gauge theory, the gauge invariance must be broken by a condensate. The condensate will then define a preferred phase, and the phase of the condensate will define the zero value of the field in a gauge-invariant way. The gauge-invariant definition is that a gauge field is zero when the phase change along any path from parallel transport is equal to the phase difference in the condensate wavefunction. The condensate value is described by a quantum field with an expectation value, just as in the Ginzburg-Landau model.
In order for the phase of the vacuum to define a gauge, the field must have a phase also referred to as 'to be charged'. The vector potential changes the phase of the quanta produced by the field when they move from point to point. In terms of fields, it defines how much to rotate the real and imaginary parts of the fields into each other when comparing field values at nearby points.
The action for this model is. The first term is the kinetic energy of the field. The second term is the extra potential energy when the field varies from point to point. The third term is the potential energy when the field has any given magnitude. This potential energy, the Higgs potential , z ,  has a graph which looks like a Mexican hat , which gives the model its name.
Starting in any one of the circle of vacua and changing the phase of the field from point to point costs very little energy. Mathematically, if. This is not a surprise. This is an example of Goldstone's theorem : spontaneously broken continuous symmetries normally produce massless excitations. The Abelian Higgs model is the Mexican-hat model coupled to electromagnetism :.
But now the phase of the field is arbitrary, because gauge transformations change it. Furthermore, choosing a gauge where the phase of the vacuum is fixed, the potential energy for fluctuations of the vector field is nonzero.
So in the Abelian Higgs model, the gauge field acquires a mass. To calculate the magnitude of the mass, consider a constant value of the vector potential A in the x -direction in the gauge where the condensate has constant phase. This is the same as a sinusoidally varying condensate in the gauge where the vector potential is zero. In the gauge where A is zero, the potential energy density in the condensate is the scalar gradient energy:.
It is exactly analogous to the Abelian Higgs model. Depending on the representation of the scalar field, not every gauge field acquires a mass. A simple example is in the renormalizable version of an early electroweak model due to Julian Schwinger. If this field has a vacuum expectation value, it points in some direction in field space. Rotations around the x and y -axis do not preserve the vacuum, and the components of the SO 3 gauge field which generate these rotations become massive vector mesons.
It doesn't break electroweak symmetry properly as in nature.
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But historically, a model similar to this but not using the Higgs mechanism was the first in which the weak force and the electromagnetic force were unified. Ernst Stueckelberg discovered  a version of the Higgs mechanism by analyzing the theory of quantum electrodynamics with a massive photon.
Effectively, Stueckelberg's model is a limit of the regular Mexican hat Abelian Higgs model, where the vacuum expectation value H goes to infinity and the charge of the Higgs field goes to zero in such a way that their product stays fixed. The mass of the Higgs boson is proportional to H , so the Higgs boson becomes infinitely massive and decouples, so is not present in the discussion. The interpretation is that when a U 1 gauge field does not require quantized charges, it is possible to keep only the angular part of the Higgs oscillations, and discard the radial part.
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To have arbitrarily small charges requires that the U 1 is not the circle of unit complex numbers under multiplication, but the real numbers R under addition, which is only different in the global topology. Such a U 1 group is non-compact. Among the allowed gauge groups, only non-compact U 1 admits affine representations, and the U 1 of electromagnetism is experimentally known to be compact, since charge quantization holds to extremely high accuracy. The Higgs condensate in this model has infinitesimal charge, so interactions with the Higgs boson do not violate charge conservation.
The theory of quantum electrodynamics with a massive photon is still a renormalizable theory, one in which electric charge is still conserved, but magnetic monopoles are not allowed. For non-Abelian gauge theory, there is no affine limit, and the Higgs oscillations cannot be too much more massive than the vectors. From Wikipedia, the free encyclopedia. Standard Model of particle physics Elementary particles of the Standard Model.
The story of the Higgs boson: the origin of mass in early particle physics
Feynman diagram. Topics to be covered at the Congress include a wide range of fundamental research areas, as well as physics education, industrial physics and applications, women in physics. There are two VIP tickets and two student tickets available in each of these states, but only for AIP associates, members and fellows.
The Bragg Medal has a separate nomination process, with entries submitted by Australian universities to their state AIP branch. The scholarship will pay tribute to four CSIRO scientists who were influential in bringing novel techniques and products to industry, by giving a post-graduate physicist an invaluable opportunity to further their own research with travel to an overseas institution. Contributions have already been received from the Laboratories Credit Union and the family, friends, colleagues and associates of the four physicists, and now further support is being sought to ensure this scholarship can be awarded annually.
Universities Australia, the peak body for our university sector, is seeking your support to send the message that investing in universities is investing in our future. For delegates though it was also a chance to learn more about the intricacies of policy-making, as well as how to get yourself heard in Canberra, in the media and in public. The winners and 14 highly commended entries will also be featured in an academic calendar for Tue 29 Apr , 5. Sir Peter Knight from the Kavli Royal Society International Centre will discuss the significance of the networked world and the quantum technology with even more dramatic potential.
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