3Discrete symmetries
III The Standard Model
3.1 Symmetry operators
How do these transformations of the universe relate to our quantum-mechanical
theories? In quantum mechanics, we have a state space
H
, and the fields are
operators
φ
(
t, x
) :
H → H
for each (
t, x
)
∈ R
1,3
. Unfortunately, we do not have
a general theory of how we can turn a classical theory into a quantum one, so
we need to make some (hopefully natural) assumptions about how this process
behaves.
The first assumption is absolutely crucial to get ourselves going.
Assumption.
Any classical transformation of the universe (e.g. P or T) gives
rise to some function f : H → H.
In general
f
need not be a linear map. However, it turns out if the transfor-
mation is in fact a symmetry of the theory, then Wigner’s theorem forces a lot
of structure on f.
Roughly, Wigner’s theorem says the following — if a function
f
:
H → H
is
a symmetry, then it is linear and unitary, or anti-linear and anti-unitary.
Definition
(Linear and anti-linear map)
.
Let
H
be a Hilbert space. A function
f : H → H is linear if
f(αΦ + βΨ) = αf(Φ) + βf(Ψ)
for all α, β ∈ C and Φ, Ψ ∈ H. A map is anti-linear if
f(αΦ + βΨ) = α
∗
f(Φ) + β
∗
f(Ψ).
Definition
(Unitary and anti-unitary map)
.
Let
H
be a Hilbert space, and
f : H → H a linear map. Then f is unitary if
hfΦ, fΨi = hΦ, Ψi
for all Φ, Ψ ∈ H.
If f : H → H is anti-linear, then it is anti-unitary if
hfΦ, fΨi = hΦ, Ψi
∗
.
But to state the theorem precisely, we need to be a bit more careful. In
quantum mechanics, we consider two (normalized) states to be equivalent if they
differ by a phase. Thus, we need the following extra assumption on our function
f:
Assumption.
In the previous assumption, we further assume that if Φ
,
Ψ
∈ H
differ by a phase, then f Φ and fΨ differ by a phase.
Now what does it mean for something to “preserve physics”? In quantum
mechanics, the physical properties are obtained by taking inner products. So
we want to require f to preserve inner products. But this is not quite what we
want, because states are only defined up to a phase. So we should only care
about inner products defined up to a phase. Note that this in particular implies
f preserves normalization.
Thus, we can state Wigner’s theorem as follows:
Theorem
(Wigner’s theorem)
.
Let
H
be a Hilbert space, and
f
:
H → H
be a
bijection such that
– If Φ, Ψ ∈ H differ by a phase, then fΦ and fΨ differ by a phase.
– For any Φ, Ψ ∈ H, we have
|hfΦ, fΨi| = |hΦ, Ψi|.
Then there exists a map
W
:
H → H
that is either linear and unitary, or
anti-linear and anti-unitary, such that for all Φ
∈ H
, we have that
W
Φ and
f
Φ
differ by a phase.
For all practical purposes, this means we can assume the transformations C,
P and T are given by unitary or anti-unitary operators.
We will write
ˆ
C
,
ˆ
P
and
ˆ
T
for the (anti-)unitary transformations induced on
the Hilbert space by the C, P, T transformations. We also write
W
(Λ
, a
) for the
induced transformation from the (not necessarily proper) Poincar´e transformation
x
µ
7→ x
0µ
= Λ
µ
ν
x
ν
+ a
µ
.
We want to understand which are unitary and which are anti-unitary. We will
assume that these W(Λ, a) “compose properly”, i.e. that
W (Λ
2
, a
2
)W (Λ
1
, a
1
) = W (Λ
2
Λ
1
, Λ
2
a
1
+ a
2
).
Moreover, we assume that for infinitesimal transformations
Λ
µ
ν
= δ
µ
ν
+ ω
µ
ν
, a
µ
= ε
µ
,
where ω and ε are small parameters, we can expand
W = W (Λ, a) = W (I + ω, ε) = 1 +
i
2
ω
µν
J
µν
+ iε
µ
P
µ
,
where
J
µν
are the operators generating rotations and boosts, and
P
µ
are the
operators generating translations. In particular, P
0
is the Hamiltonian.
Of course, we cannot write parity and time reversal in this form, because
they are discrete symmetries, but we can look at what happens when we combine
these transformations with infinitesimal ones.
By assumption, we have
ˆ
P = W (P, 0),
ˆ
T = W (T, 0).
Then from the composition rule, we expect
ˆ
P W
ˆ
P
−1
= W (PΛP
−1
, Pa)
ˆ
T W
ˆ
T
−1
= W (TΛT
−1
, Ta).
Inserting expansions for
W
in terms of
ω
and
ε
on both sides, and comparing
coefficients of ε
0
, we find
ˆ
P iH
ˆ
P
−1
= iH
ˆ
T iH
ˆ
T
−1
= −iH.
So iH and
ˆ
P commute, but iH and
ˆ
T anti-commute.
To proceed further, we need to make the following assumption, which is a
natural one to make if we believe P and T are symmetries.
Assumption.
The transformations
ˆ
P
and
ˆ
T
send an energy eigenstate of energy
E to an energy eigenstate of energy E.
From this, it is easy to figure out whether the maps
ˆ
P
and
ˆ
T
should be
unitary or anti-unitary.
Indeed, consider any normalized energy eigenstate Ψ with energy
E 6
= 0.
Then by definition, we have
hΨ, iHΨi = iE.
Then since
ˆ
P Ψ is also an energy eigenstate of energy E, we know
iE = h
ˆ
P Ψ, iH
ˆ
P Ψi = h
ˆ
P Ψ,
ˆ
P iHΨi.
In other words, we have
h
ˆ
P Ψ,
ˆ
P iHΨi = hΨ, iHΨi = iE.
So
ˆ
P must be unitary.
On the other hand, we have
iE = h
ˆ
T Ψ, iH
ˆ
T Ψi = −h
ˆ
T Ψ,
ˆ
T iHΨi.
In other words, we obtain
h
ˆ
T Ψ,
ˆ
T iHΨi = −hΨ, iHΨi = hΨ, iHΨi
∗
− iE.
Thus, it follows that
ˆ
T
must be anti-unitary. It is a fact that
ˆ
C
is linear and
unitary.
Note that these derivations rely crucially on the fact that we know the
operators must either by unitary or anti-unitary, and this allows us to just check
one inner product to determine which is the case, rather than all.
So far, we have been discussing how these operators act on the elements of
the state space. However, we are ultimately interested in understanding how
the fields transform under these transformations. From linear algebra, we know
that an endomorphism
W
:
H → H
on a vector space canonically induces a
transformation on the space of operators, by sending
φ
to
W φW
−1
. Thus, what
we want to figure out is how W φW
−1
relates to φ.