One troubling aspect of spontaneously broken gauge
field theories based on the Higgs mechanism of giving mass to gauge
bosons is that it's not only the coupling constants, but also the masses,
that get renormalized by quantum corrections from taking into account
all possible virtual processes at all possible momentum scales.
Suppose there is new physics at some scale L,
so that the Standard Model of particle physics is no longer adequate
to describe physics at higher momentum scales. The quantum corrections
to fermion masses would depend on that cutoff scale L
only logarithmically
whereas the scalar Higgs particles would exhibit a quadratic dependence
on the cutoff scale
This means that the masses of Higgs particles are very sensitive to
the scale at which new physics emerges.
This sensitivity is called the gauge
hierarchy problem, because the Higgs mass is related to the masses
of the gauge bosons in the spontaneously broken gauge theory. The original
question "How do the gauge bosons get mass
without spoiling gauge invariance?" was only partially answered
by the Higgs mechanism. In a way, question wasn't answered by the Higgs
mechanism, it was just transferred up to a new level, to the question:
"Why does the Higgs mass remain stable against
large quantum corrections from high energy scales?"
The interesting thing about scalar
mass divergences from virtual particle loops is that virtual
fermions and virtual bosons contribute with opposite signs and could
cancel each other completely if for
every boson, there were a fermion of the same mass and charge.
At the level of quantum mechanics, this type of FermiBose
symmetry would entail some quantum operator, let's call it Q, whose
action would be to transform bosons into fermions, and vice versa. In
operator language this would be written
And since this is a symmetry, this operator must commute with the Hamiltonian
Such a theory is called a supersymmetric theory,
and the operator Q is called the supercharge.
Since the supercharge corresponds to an operator that changes a particle
with spin one half to a particle with spin one or zero, the supercharge
itself must be a spinor that carries one half unit of spin of its own.
Supersymmetry is such a powerful idea because it is
a symmetry under the exchange of classical and quantum physics. Bosons
are particles that obey Bose statistics, meaning that any number of
them can occupy the same quantum state at the same time. Fermions obey
Fermi statistics and only one fermion can occupy any given quantum state
at one time. But the classical limit of quantum physics is approached
when the occupation numbers of available states are very high. For example,
in this limit, the quantum photon field behaves
like the classical electromagnetic field as described by Maxwell's
equations. But then the conclusion for fermions is that there
is no classical limit for fermions. Fermionic fields are inherently
quantum relativistic phenomena.
Therefore, any symmetry that exchanges fermions and
bosons is a symmetry that exchanges physics
that has a classical limit with physics that has no classical limit.
So such a symmetry should have very powerful consequences indeed.
One big problem with supersymmetry: in the particle
physics that is observed in today's accelerators, every boson
most definitely does NOT have a matching fermion with the same
mass and charge. So if supersymmetry is a symmetry of Nature, it
must somehow be broken. It's easy enough for an expert to construct
a supersymmetric theory. It's breaking the symmetry, without destroying
the beneficial effects of that symmetry, that has been the hardest part
of the program to fulfill.
But would a broken supersymmetric theory still be
able to solve the gauge hierarchy problem? That depends on the scale
at which the supersymmetry is broken, and the method by which it is
broken. In other words, it's still an open question. Stay tuned.
How was supersymmetry developed?
Supersymmetry was not developed originally as a means
of solving the gauge hierarchy problem. Supersymmetry was first developed
independently by two different groups of theorists separated by the
Cold War back in the 1970s. One group in the USSR was exploring the
mathematics of spacetime symmetry, and the other group in the West
was trying to add fermions to bosonic string theory.
In the USSR, mathematicians Gol'fand and Likhtman
wanted to do something exotic with the group theory of spacetime symmetries.
The usual group of spacetime symmetries in relativistic quantum field
theory is called the Poincaré group. This group includes symmetries
under spatial rotations, spacetime boosts and translations in space
and time.
The action of the group can be described by the algebra
of the group, which is defined by a set of
commutation relations between the generators of infinitesimal
group transformations. The algebra of the Poincaré group looks
like:
The momentum generator P^{m}
generates space and time translations. The Lorentz matrices J_{mn}
generate rotations in space and Lorentz boosts in spacetime. These are
all bosonic symmetries, which ought to be true because momentum conservation
and Lorentz invariance are present in classical physics.
But the Poincaré group also has representations
that describe fermions. Since spin 1/2
particles arise as solutions to a relativistically invariant equation
 the Dirac equation  this is to be expected. If there are spin 1/2
particles, could there be spin 1/2 symmetry generators in a spacetime
symmetry algebra? Yes! One way to add them is shown below:
What are the new symmetry generators labeled by Q? These are the supercharges
mentioned above.
What Gol'fand and Likhtman ended up with was the
group theory of supersymmetric transformations
in four spacetime dimensions, and using this new type of symmetry,
they constructed the first supersymmetric quantum
field theory.
Unfortunately for them, their work was ignored, both
in the Soviet Union and in the West, until years later when supersymmetry
finally mushroomed into a major topic of investigation in particle physics.
In 1972, Gol'fand was judged one of the least important researchers
in his group at FIAN in Moscow, and so he was let go in a cost reduction
drive in 1973. He remained unemployed for seven years, until pressure
from the world physics community led to his rehiring in 1980.
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