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Generators of a Supergroup
--------------------------
Creation and Annihilation Operators
-----------------------------------
Supersymmetry is a symmetry that interchanges bosons
with fermions and vice versa.
The creation and annihilation operators of bosons and
fermions obey the following commutation rules:
Bosons:
[a_{i}^{-},a_{j}^{+}] = δ_{ij}
[a_{i}^{+},a_{j}^{+}] = [a_{i}^{-},a_{j}^{-}] = 0
Fermions:
{c_{i}^{-},c_{j}^{+}} = δ_{ij}
{c_{i}^{+},c_{j}^{+}} = {c_{i}^{-},c_{j}^{-}} = 0 for i = j
This is reflective of the Pauli Exclusiom Principle that
disallows 2 or more fermions to bein thesame state.
Both together:
[a^{+},c^{+}] = [a^{-},c^{-}] = [a^{+},c^{-}] = [a^{-},c^{+}] = 0
Interchange of 2 bosons:
[a_{i}^{+},a_{j}^{+}] = a_{i}^{+}a_{j}^{+} - a_{j}^{+}a_{i}^{+} = 0 => a_{i}^{+}a_{j}^{+} = a_{j}^{+}a_{i}^{+}
Therefore,
a_{i}^{+}a_{j}^{+}|0> = a_{j}^{+}a_{i}^{+}|0>
Interchange of 2 fermions:
{c_{i}^{+},c_{j}^{+}} = c_{i}^{+}c_{j}^{+} + c_{j}^{+}c_{i}^{+} = 0 => c_{i}^{+}c_{j}^{+} = -c_{j}^{+}c_{i}^{+}
Therefore,
c_{i}^{+}c_{j}^{+}|0> = -c_{j}^{+}c_{i}^{+}|0>
We can see this diagramatically as follows:
Boson: φ(x_{A},x_{B}) = φ(x_{B},x_{A}) - symmetric
[φ(x_{A}),φ(x_{B})] = [φ(x_{B}),φ(x_{A})]
Fermion: ψ(x_{A},x_{B}) = -ψ(x_{B},x_{A}) - antisymmetric
{ψ(x_{A}),ψ(x_{B})} = -{φ(x_{B}),φ(x_{A})}
{θ_{A},θ_{B}} = -{θ_{B},θ_{A}}
Grassmann Algebra
-----------------
There is no limit to number of bosons in a state.
Therefore, the field operators can grow large. On the
other hand, the Pauli Exclusion Principle prohibits 2
or more fermions from occupying the same state because
(c^{+})^{2} = (c^{-})^{2} = 0. Therefore, the behavior of these
fields is encoded in the commutator and anti-commutator
relationships respectively. There is a special algebra
that describes the anti-commuting properties of fermionic
fields called GRASSMANN ALGEBRA. In Grassmann algebra,
ordinary numbers that commute are replaced by numbers
that anti-commute. These anti-commuting numbers are
extremely useful in keeping track of fermionic fields.
In essence, Fermionic fields are really Grassmann numbers.
COMMUTING NUMBERS ARE REFERRED TO AS EVEN ELEMENTS OF
THE GRASSMANN ALGEBRA AND CORRESPOND TO DIRECTLY
MEASURABLE QUANTITIES. ANTI-COMMUTING NUMBERS ARE
REFERRED TO AS ODD ELEMENTS OF THE GRASSMANN ALGEBRA
AND CORRESPOND TO QUANTUM MECHANICAL PROBABILITIES.
BOSONIC FIELDS ARE EVEN ELEMENTS AND FERMIONIC FIELDS
ARE ODD ELEMENTS.
Generators of a Supergroup, Q
-----------------------------
Lie Algebra
-----------
The Lie algebra of the infinitessimal generators for
even elements, G, is the familiar:
[G_{i},G_{j}] = iε_{ijk}G_{k}
In Supersymmetry, we for look the infinitessimal group
generators that transform a bosonic field to a fermionic
field and vice versa.
The generators are defined as:
Q = (√m)a^{+}c^{-} ... removes a fermion and adds a boson
Q^{†} = (√m)a^{-}c^{+} ... adds a fermion and remove a boson
Because they are composed of the products a^{+}c^{-} or a^{-}c^{+},
which are even and odd elements respectively, these
generators represent the odd elements of the Grassmann
algebra. Therefore, they follow the same anti-
commutation rules as the creation and annihilation
operators for fermions.
Therefore,
{Q,Q} = m(a^{+}c^{-}a^{+}c^{-}) = 0 since (c^{-})^{2} = 0.
Likewise, {Q^{†},Q^{†}} = 0
It is important to note that the Q and Q^{†} operators
act on a SINGLE particle. Because QQ = 0, the operations
cannot be compounded in the same way that other generators
can (i.e. generators of rotation). In this sense they are
different from other generators that we have encountered.
The other anti-commutations are:
{Q^{†},Q^{†}} = m(a^{-}c^{+}a^{-}c^{+}) = 0 since (c^{+})^{2} = 0
{Q,Q^{†}} = m(a^{+}c^{-}a^{-}c^{+} + a^{-}c^{+}a^{+}c^{-})
^{ } = m(a^{+}a^{-}c^{-}c^{+} + a^{-}a^{+}c^{+}c^{-} + a^{+}a^{-}c^{+}c^{-} - a^{+}a^{-}c^{+}c^{-})
^{ } = m(a^{+}a^{-} + c^{+}c^{-})
^{ } = m(n_{B} + n_{F})
^{ } = 2H ... assumes boson mass = fermion mass
^{ } = 2i∂/i∂t
^{ } = 2p_{0}
p_{0} is the energy component of p_{μ} which we will encounter
when we extend the discussion to spacetime.
From the rules for Grassmann algebra, a mix of Grassmann
(odd elements) and ordinary numbers (even elements) are
always COMMUTED. H is an even element. The commutator
of the Q's with H are:
[Q,H] = [Q^{†},H] = 0
Proof:
H|ψ> = E|ψ>
Q = Symmetry operation (rotation/translation)
HQ|ψ> = EQ|ψ>
= QE|ψ>
= QH|ψ>
(HQ - QH)|ψ> = 0
[Q,H] = -[H,Q] = 0
This means that Q and Q^{†} are both conserved quantities.
Therefore, internal symmetries that take one particle to
another involve time translations. In the Standard Model,
for example, isospin symmetries take protons to neutrons,
and color symmetries take red quarks to blue quarks but
neither of these symmetries involves spacetime. This is
something quite new!
There is another way to think about this. Instead of
time translation, we can regard the supersymmetry
generators as shifts in the coordinates which are
Grassmann numbers (Grassmann coordinates).
Think about a superfield, Φ, defined in terms of Grassmann
_ _
numbers, θ, θ, and time, t, Φ(θ,θ,t). A simple field
like this may look like:
_ _ _
Φ(t) = φ(t) + θψ(t) + ψ(t)θ + D(t)θθ
E = B O = F O = F E = B
Consider the following symmetry transformations:
θ -> θ + ξ ... spatial translation
_ _ _
θ -> θ + ξ ... spatial translation
_ _
t -> t + iξθ + iξθ ... time translation invariance.
Where ξ is a infinitesimally small Grassmann number.
(Note: ξ^{2} = 0 so it behave like ε in Group Theory).
The change in Φ is given by:
_ _ _ _ _
δΦ(θ,θ,t) = ξ∂Φ/∂θ + ξ∂Φ/∂θ + iξθ∂Φ/∂t + iξθ∂Φ/∂t
_ _
= ξ(∂/∂θ + iθ∂/∂t)Φ + ξ[...]Φ
_
= (ξQ + ξQ^{†})Φ
Where,
_
Q^{†} = ∂/∂θ + iθ∂/∂t
and
_
Q = ∂/∂θ + iθ∂/∂t
Therefore,
_ _
{Q,Q^{†}} = {∂/∂θ + iθ∂/∂t,∂/∂θ + iθ∂/∂t}
Making use of the following properties of Grassmann
numbers:
_
(∂/∂θ)(∂/∂θ) = θθ^{†} = 0
_ _
(∂/∂θ)θ = (∂/∂θ)θ = 1
Leaves us with,
_ _
{Q,Q^{†}} = i(∂/∂θ)θ(∂/∂t) + i(∂/∂θ)θ(∂/∂t)
{Q,Q^{†}} = -2i∂/∂t as we calculated before.
So there is alternative to the creation and annihilation
formulation of the Q's that involves differential operators
that involves a new type of spacetime involving anti-commuting
coordinates. They generate energy in the ordinary coordinates
_
via the iθ∂/∂t term, but also perform spatial transformations
in the Grassmann coordinates via the ∂/∂θ term.