WeinbergSalemModel - crowlogic/arb4j GitHub Wiki

The quantum field and group theoretical aspects of the Weinberg-Salam model are very important. Here is an outline of the essential ideas:

The electroweak theory is a quantum field theory, which means that it treats particles as excited states of an underlying quantum field. The quantum field for each type of particle (such as an electron, a photon, or a quark) can be thought of as permeating all of space and time. When the field is excited, a particle is created.

The theory is also a gauge theory, a type of field theory that includes a symmetry principle, or gauge symmetry. The particular gauge symmetry in the electroweak theory is based on the group SU(2) x U(1).

The group SU(2) represents the weak isospin, a kind of quantum number related to the weak interaction, and U(1) represents weak hypercharge, another quantum number. A fundamental aspect of the Weinberg-Salam model is that the electromagnetic charge of a particle can be calculated from its weak isospin and weak hypercharge.

When the electroweak theory is formulated mathematically, one of the consequences of the gauge symmetry is that the gauge bosons (the W+, W-, and Z bosons for SU(2) and the B boson for U(1)) should be massless. However, this contradicts experimental observation, which shows that the W and Z bosons are quite heavy.

The solution to this discrepancy is the Higgs mechanism, which involves introducing an additional field, the Higgs field, and allowing it to interact with the gauge fields. The Higgs field has a non-zero vacuum expectation value, which gives the appearance of spontaneous symmetry breaking, allowing the W and Z bosons to acquire mass while keeping the photon massless.

However, the gauge symmetry is not truly broken – it is merely hidden or obscured, as you suggested. From a more abstract, mathematical perspective, the gauge symmetry is always preserved, but the vacuum state doesn't respect this symmetry, a concept known as "spontaneous symmetry breaking."

Also, in quantum field theory, it is often more convenient to work with so-called "renormalizable" theories, ones that can be consistently calculated to arbitrarily high energy. The Higgs mechanism, along with gauge invariance, ensures that the Standard Model, including the electroweak theory, is renormalizable, making it a powerful tool for calculations in high-energy physics.