In quantum mechanics, a parity transformation
(also called parity inversion) is the flip
in the sign of one spatial coordinate. In
three dimensions, it can also refer to the
simultaneous flip in the sign of all three
spatial coordinates (a point reflection):
P
:
(
x
y
z
)
↦
(
−
x
−
y
−
z
)
.
{\displaystyle \mathbf {P} :{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto
{\begin{pmatrix}-x\\-y\\-z\end{pmatrix}}.}
It can also be thought of as a test for chirality
of a physical phenomenon, in that a parity
inversion transforms a phenomenon into its
mirror image. All fundamental interactions
of elementary particles, with the exception
of the weak interaction, are symmetric under
parity. The weak interaction is chiral and
thus provides a means for probing chirality
in physics. In interactions that are symmetric
under parity, such as electromagnetism in
atomic and molecular physics, parity serves
as a powerful controlling principle underlying
quantum transitions.
A matrix representation of P (in any number
of dimensions) has determinant equal to −1,
and hence is distinct from a rotation, which
has a determinant equal to 1. In a two-dimensional
plane, a simultaneous flip of all coordinates
in sign is not a parity transformation; it
is the same as a 180°-rotation.
In quantum mechanics, wave functions which
are unchanged by a parity transformation are
described as even functions, while those which
change sign under a parity transformation
are odd functions.
== Simple symmetry relations ==
Under rotations, classical geometrical objects
can be classified into scalars, vectors, and
tensors of higher rank. In classical physics,
physical configurations need to transform
under representations of every symmetry group.
Quantum theory predicts that states in a Hilbert
space do not need to transform under representations
of the group of rotations, but only under
projective representations. The word projective
refers to the fact that if one projects out
the phase of each state, where we recall that
the overall phase of a quantum state is not
an observable, then a projective representation
reduces to an ordinary representation. All
representations are also projective representations,
but the converse is not true, therefore the
projective representation condition on quantum
states is weaker than the representation condition
on classical states.
The projective representations of any group
are isomorphic to the ordinary representations
of a central extension of the group. For example,
projective representations of the 3-dimensional
rotation group, which is the special orthogonal
group SO(3), are ordinary representations
of the special unitary group SU(2) (see Representation
theory of SU(2)). Projective representations
of the rotation group that are not representations
are called spinors, and so quantum states
may transform not only as tensors but also
as spinors.
If one adds to this a classification by parity,
these can be extended, for example, into notions
of
scalars (P = +1) and pseudoscalars (P = −1)
which are rotationally invariant.
vectors (P = −1) and axial vectors (also
called pseudovectors) (P = +1) which both
transform as vectors under rotation.One can
define reflections such as
V
x
:
(
x
y
z
)
↦
(
−
x
y
z
)
,
{\displaystyle V_{x}:{\begin{pmatrix}x\\y\\z\end{pmatrix}}\mapsto
{\begin{pmatrix}-x\\y\\z\end{pmatrix}},}
which also have negative determinant and form
a valid parity transformation. Then, combining
them with rotations (or successively performing
x-, y-, and z-reflections) one can recover
the particular parity transformation defined
earlier. The first parity transformation given
does not work in an even number of dimensions,
though, because it results in a positive determinant.
In odd number of dimensions only the latter
example of a parity transformation (or any
reflection of an odd number of coordinates)
can be used.
Parity forms the abelian group
Z
2
{\displaystyle \mathbb {Z} _{2}}
due to the relation
P
^
2
=
1
^
{\displaystyle {\hat {\mathcal {P}}}^{2}={\hat
{1}}}
. All Abelian groups have only one-dimensional
irreducible representations. For
Z
2
{\displaystyle \mathbb {Z} _{2}}
, there are two irreducible representations:
one is even under parity,
P
^
ϕ
=
+
ϕ
{\displaystyle {\hat {\mathcal {P}}}\phi =+\phi
}
, the other is odd,
P
^
ϕ
=
−
ϕ
{\displaystyle {\hat {\mathcal {P}}}\phi =-\phi
}
. These are useful in quantum mechanics. However,
as is elaborated below, in quantum mechanics
states need not transform under actual representations
of parity but only under projective representations
and so in principle a parity transformation
may rotate a state by any phase.
== Classical mechanics ==
Newton's equation of motion
F
→
=
m
a
→
{\displaystyle {\vec {F}}=m\,{\vec {a}}}
(if the mass is constant) equates two vectors,
and hence is invariant under parity. The law
of gravity also involves only vectors and
is also, therefore, invariant under parity.
However, angular momentum
L
→
{\displaystyle {\vec {L}}}
is an axial vector,
L
→
=
r
→
×
p
→
{\displaystyle {\vec {L}}={\vec {r}}\times
{\vec {p}}}
,
P
^
(
L
→
)
=
−
r
→
×
−
p
→
=
L
→
{\displaystyle {\hat {P}}\left({\vec {L}}\right)=-\,{\vec
{r}}\times -\,{\vec {p}}={\vec {L}}}
.In classical electrodynamics, the charge
density
ρ
{\displaystyle \rho }
is a scalar, the electric field,
E
→
{\displaystyle {\vec {E}}}
, and current
j
→
{\displaystyle {\vec {j}}}
are vectors, but the magnetic field,
H
→
{\displaystyle {\vec {H}}}
is an axial vector. However, Maxwell's equations
are invariant under parity because the curl
of an axial vector is a vector.
== Effect of spatial inversion on some variables
of classical physics ==
=== Even ===
Classical variables, predominantly scalar
quantities, which do not change upon spatial
inversion include:
t
{\displaystyle \ t}
, the time when an event occurs
m
{\displaystyle \ m}
, the mass of a particle
E
{\displaystyle \ E}
, the energy of the particle
P
{\displaystyle \ P}
, power (rate of work done)
ρ
{\displaystyle \ \rho }
, the electric charge density
V
{\displaystyle \ V}
, the electric potential (voltage)
ρ
{\displaystyle \ \rho }
, energy density of the electromagnetic field
L
{\displaystyle \mathbf {L} }
, the angular momentum of a particle (both
orbital and spin) (axial vector)
B
{\displaystyle \mathbf {B} }
, the magnetic field (axial vector)
H
{\displaystyle \mathbf {H} }
, the auxiliary magnetic field
M
{\displaystyle \mathbf {M} }
, the magnetization
T
i
j
{\displaystyle \ T_{ij}}
Maxwell stress tensor.
All masses, charges, coupling constants, and
other physical constants, except those associated
with the weak force
=== Odd ===
Classical variables, predominantly vector
quantities, which have their sign flipped
by spatial inversion include:
h
{\displaystyle \ h}
, the helicity
Φ
{\displaystyle \ \Phi }
, the magnetic flux
x
{\displaystyle \mathbf {x} }
, the position of a particle in three-space
v
{\displaystyle \mathbf {v} }
, the velocity of a particle
a
{\displaystyle \mathbf {a} }
, the acceleration of the particle
p
{\displaystyle \mathbf {p} }
, the linear momentum of a particle
F
{\displaystyle \mathbf {F} }
, the force exerted on a particle
J
{\displaystyle \mathbf {J} }
, the electric current density
E
{\displaystyle \mathbf {E} }
, the electric field
D
{\displaystyle \mathbf {D} }
, the electric displacement field
P
{\displaystyle \mathbf {P} }
, the electric polarization
A
{\displaystyle \mathbf {A} }
, the electromagnetic vector potential
S
{\displaystyle \mathbf {S} }
, Poynting vector.
== Quantum mechanics ==
=== Possible eigenvalues ===
In quantum mechanics, spacetime transformations
act on quantum states. The parity transformation,
P
^
{\displaystyle {\hat {\mathcal {P}}}}
, is a unitary operator, in general acting
on a state
ψ
{\displaystyle \psi }
as follows:
P
^
ψ
(
r
)
=
e
i
ϕ
2
ψ
(
−
r
)
{\displaystyle {\hat {\mathcal {P}}}\,\psi
_{\left(r\right)}=e^{\frac {i\phi }{2}}\psi
_{\left(-r\right)}}
.
One must then have
P
^
2
ψ
(
r
)
=
e
i
ϕ
ψ
(
r
)
{\displaystyle {\hat {\mathcal {P}}}^{2}\,\psi
_{\left(r\right)}=e^{i\phi }\psi _{\left(r\right)}}
, since an overall phase is unobservable.
The operator
P
^
2
{\displaystyle {\hat {\mathcal {P}}}^{2}}
, which reverses the parity of a state twice,
leaves the spacetime invariant, and so is
an internal symmetry which rotates its eigenstates
by phases
e
i
ϕ
{\displaystyle e^{i\phi }}
. If
P
^
2
{\displaystyle {\hat {\mathcal {P}}}^{2}}
is an element
e
i
Q
{\displaystyle e^{iQ}}
of a continuous U(1) symmetry group of phase
rotations, then
e
−
i
Q
{\displaystyle e^{-iQ}}
is part of this U(1) and so is also a symmetry.
In particular, we can define
P
^
′
≡
P
^
e
−
i
Q
2
{\displaystyle {\hat {\mathcal {P}}}'\equiv
{\hat {\mathcal {P}}}\,e^{-{\frac {iQ}{2}}}}
, which is also a symmetry, and so we can
choose to call
P
^
′
{\displaystyle {\hat {\mathcal {P}}}'}
our parity operator, instead of
P
^
2
{\displaystyle {\hat {\mathcal {P}}}^{2}}
. Note that
P
^
′
2
=
1
{\displaystyle {{\hat {\mathcal {P}}}'}^{2}=1}
and so
P
^
′
{\displaystyle {\hat {\mathcal {P}}}'}
has eigenvalues
±
1
{\displaystyle \pm 1}
. Wave functions with eigenvalue +1 under
a parity transformation are even functions,
while eigenvalue -1 corresponds to odd functions.
However, when no such symmetry group exists,
it may be that all parity transformations
have some eigenvalues which are phases other
than
±
1
{\displaystyle \pm 1}
.
For electronic wavefunctions, even states
are usually indicated by a subscript g for
gerade (German: even) and odd states by a
subscript u for ungerade (German: odd). For
example, the lowest energy level of the hydrogen
molecule ion (H2+) is labelled
1
σ
g
{\displaystyle 1\sigma _{g}}
and the next-closest (higher) energy level
is labelled
1
σ
u
{\displaystyle 1\sigma _{u}}
.The wave functions of a particle moving into
an external potential, which is centrosymmetric
(potential energy invariant with respect to
a space inversion, symmetric to the origin),
either remain invariable or change signs:
these two possible states are called the even
state or odd state of the wave functions.The
law of conservation of parity of particle
(not true for the beta decay of nuclei) states
that, if an isolated ensemble of particles
has a definite parity, then the parity remains
invariable in the process of ensemble evolution.
The parity of the states of a particle moving
in a spherically symmetric external field
is determined by the angular momentum, and
the particle state is defined by three quantum
numbers: total energy, angular momentum and
the projection of angular momentum.
=== Consequences of parity symmetry ===
When parity generates the Abelian group ℤ2,
one can always take linear combinations of
quantum states such that they are either even
or odd under parity (see the figure). Thus
the parity of such states is ±1. The parity
of a multiparticle state is the product of
the parities of each state; in other words
parity is a multiplicative quantum number.
In quantum mechanics, Hamiltonians are invariant
(symmetric) under a parity transformation
if
P
^
{\displaystyle {\hat {\mathcal {P}}}}
commutes with the Hamiltonian. In non-relativistic
quantum mechanics, this happens for any potential
which is scalar, i.e.,
V
=
V
(
r
)
{\displaystyle V=V{\left(r\right)}}
, hence the potential is spherically symmetric.
The following facts can be easily proven:
If
|
φ
⟩
{\displaystyle \left|\varphi \right\rangle
}
and
|
ψ
⟩
{\displaystyle \left|\psi \right\rangle }
have the same parity, then
⟨
φ
|
X
^
|
ψ
⟩
=
0
{\displaystyle \left\langle \varphi \right|{\hat
{X}}\left|\psi \right\rangle =0}
where
X
^
{\displaystyle {\hat {X}}}
is the position operator.
For a state
|
L
→
,
L
z
⟩
{\displaystyle \left|{\vec {L}},L_{z}\right\rangle
}
of orbital angular momentum
L
→
{\displaystyle {\vec {L}}}
with z-axis projection
L
z
{\displaystyle L_{z}}
, then
P
^
|
L
→
,
L
z
⟩
=
(
−
1
)
L
|
L
→
,
L
z
⟩
{\displaystyle {\hat {\mathcal {P}}}\left|{\vec
{L}},L_{z}\right\rangle =\left(-1\right)^{L}\left|{\vec
{L}},L_{z}\right\rangle }
.
If
[
H
^
,
P
^
]
=
0
{\displaystyle \left[{\hat {H}},{\hat {P}}\right]=0}
, then atomic dipole transitions only occur
between states of opposite parity.
If
[
H
^
,
P
^
]
=
0
{\displaystyle \left[{\hat {H}},{\hat {P}}\right]=0}
, then a non-degenerate eigenstate of
H
^
{\displaystyle {\hat {H}}}
is also an eigenstate of the parity operator;
i.e., a non-degenerate eigenfunction of
H
^
{\displaystyle {\hat {H}}}
is either invariant to
P
^
{\displaystyle {\hat {\mathcal {P}}}}
or is changed in sign by
P
^
{\displaystyle {\hat {\mathcal {P}}}}
.Some of the non-degenerate eigenfunctions
of
H
^
{\displaystyle {\hat {H}}}
are unaffected (invariant) by parity
P
^
{\displaystyle {\hat {\mathcal {P}}}}
and the others will be merely reversed in
sign when the Hamiltonian operator and the
parity operator commute:
P
^
|
ψ
⟩
=
c
|
ψ
⟩
{\displaystyle {\hat {\mathcal {P}}}\left|\psi
\right\rangle =c\left|\psi \right\rangle }
,where
c
{\displaystyle c}
is a constant, the eigenvalue of
P
^
{\displaystyle {\hat {\mathcal {P}}}}
,
P
^
2
|
ψ
⟩
=
c
P
^
|
ψ
⟩
{\displaystyle {\hat {\mathcal {P}}}^{2}\left|\psi
\right\rangle =c\,{\hat {\mathcal {P}}}\left|\psi
\right\rangle }
.
== Many-particle systems: atoms, molecules,
nuclei ==
The overall parity of a many-particle system
is the product of the parities of the one-particle
states. It is -1 if an odd number of particles
are in odd-parity states, and +1 otherwise.
Different notations are in use to denote the
parity of nuclei, atoms, and molecules.
=== Atoms ===
Atomic orbitals have parity (-1)ℓ, where
the exponent ℓ is the azimuthal quantum
number. The parity is odd for orbitals p,
f, ... with ℓ = 1, 3, ..., and an atomic
state has odd parity if an odd number of electrons
occupy these orbitals. For example, the ground
state of the nitrogen atom has the electron
configuration 1s22s22p3, and is identified
by the term symbol 4So, where the superscript
o denotes odd parity. However the third excited
term at about 83,300 cm−1 above the ground
state has electron configuration 1s22s22p23s
has even parity since there are only two 2p
electrons, and its term symbol is 4P (without
an o superscript).
=== Molecules ===
Only some molecules have a centre of symmetry,
including all homonuclear diatomic molecules
as well as certain symmetric molecules including
ethylene, benzene, xenon tetrafluoride and
sulphur hexafluoride. For such centrosymmetric
molecules, the parity each molecular orbital
is either g (gerade or even) or u (ungerade
or odd). An electronic state is u if and only
if it contains an odd number of electrons
in u orbitals.
For molecules with no centre of symmetry,
including all heteronuclear diatomics as well
as the majority of polyatomics, inversion
is not a symmetry operation and the orbitals
and states cannot be described as even or
odd.
=== Nuclei ===
In atomic nuclei, the state of each nucleon
(proton or neutron) has even or odd parity,
and nucleon configurations can be predicted
using the nuclear shell model. As for electrons
in atoms, the nucleon state has odd overall
parity if and only if the number of nucleons
in odd-parity states is odd. The parity is
usually written as a + (even) or – (odd)
following the nuclear spin value. For example
the isotopes of oxygen include 17O(5/2+),
meaning that the spin is 5/2 and the parity
is even. The shell model explains this because
the first 16 nucleons are paired so that each
pair has spin zero and even parity, and the
last nucleon is in the 1d5/2 shell which has
even parity since ℓ = 2 for a d orbital.
== Quantum field theory ==
The intrinsic parity assignments in this section
are true for relativistic quantum mechanics
as well as quantum field theory.If we can
show that the vacuum state is invariant under
parity,
P
^
|
0
⟩
=
|
0
⟩
{\displaystyle {\hat {\mathcal {P}}}\left|0\right\rangle
=\left|0\right\rangle }
, the Hamiltonian is parity invariant
[
H
^
,
P
^
]
{\displaystyle \left[{\hat {H}},{\hat {\mathcal
{P}}}\right]}
and the quantization conditions remain unchanged
under parity, then it follows that every state
has good parity, and this parity is conserved
in any reaction.
To show that quantum electrodynamics is invariant
under parity, we have to prove that the action
is invariant and the quantization is also
invariant. For simplicity we will assume that
canonical quantization is used; the vacuum
state is then invariant under parity by construction.
The invariance of the action follows from
the classical invariance of Maxwell's equations.
The invariance of the canonical quantization
procedure can be worked out, and turns out
to depend on the transformation of the annihilation
operator:
Pa(p, ±)P+ = −a(−p, ±)where p denotes
the momentum of a photon and ± refers to
its polarization state. This is equivalent
to the statement that the photon has odd intrinsic
parity. Similarly all vector bosons can be
shown to have odd intrinsic parity, and all
axial-vectors to have even intrinsic parity.
There is a straightforward extension of these
arguments to scalar field theories which shows
that scalars have even parity, since
Pa(p)P+ = a(−p).This is true even for a
complex scalar field. (Details of spinors
are dealt with in the article on the Dirac
equation, where it is shown that fermions
and antifermions have opposite intrinsic parity.)
With fermions, there is a slight complication
because there is more than one spin group.
== Parity in the standard model ==
=== Fixing the global symmetries ===
In the Standard Model of fundamental interactions
there are precisely three global internal
U(1) symmetry groups available, with charges
equal to the baryon number B, the lepton number
L and the electric charge Q. The product of
the parity operator with any combination of
these rotations is another parity operator.
It is conventional to choose one specific
combination of these rotations to define a
standard parity operator, and other parity
operators are related to the standard one
by internal rotations. One way to fix a standard
parity operator is to assign the parities
of three particles with linearly independent
charges B, L and Q. In general one assigns
the parity of the most common massive particles,
the proton, the neutron and the electron,
to be +1.
Steven Weinberg has shown that if P2 = (−1)F,
where F is the fermion number operator, then,
since the fermion number is the sum of the
lepton number plus the baryon number, F = B
+ L, for all particles in the Standard Model
and since lepton number and baryon number
are charges Q of continuous symmetries eiQ,
it is possible to redefine the parity operator
so that P2 = 1. However, if there exist Majorana
neutrinos, which experimentalists today believe
is possible, their fermion number is equal
to one because they are neutrinos while their
baryon and lepton numbers are zero because
they are Majorana, and so (−1)F would not
be embedded in a continuous symmetry group.
Thus Majorana neutrinos would have parity
±i.
=== Parity of the pion ===
In 1954, a paper by William Chinowsky and
Jack Steinberger demonstrated that the pion
has negative parity. They studied the decay
of an "atom" made from a deuteron (21H+) and
a negatively charged pion (π−) in a state
with zero orbital angular momentum
L
=
0
{\displaystyle L=0}
into two neutrons (
n
{\displaystyle n}
).
Neutrons are fermions and so obey Fermi–Dirac
statistics, which implies that the final state
is antisymmetric. Using the fact that the
deuteron has spin one and the pion spin zero
together with the antisymmetry of the final
state they concluded that the two neutrons
must have orbital angular momentum
L
=
1
{\displaystyle L=1}
. The total parity is the product of the intrinsic
parities of the particles and the extrinsic
parity of the spherical harmonic function
(
−
1
)
L
{\displaystyle \left(-1\right)^{L}}
. Since the orbital momentum changes from
zero to one in this process, if the process
is to conserve the total parity then the products
of the intrinsic parities of the initial and
final particles must have opposite sign. A
deuteron nucleus is made from a proton and
a neutron, and so using the aforementioned
convention that protons and neutrons have
intrinsic parities equal to
+
1
{\displaystyle +1}
they argued that the parity of the pion is
equal to minus the product of the parities
of the two neutrons divided by that of the
proton and neutron in the deuteron, explicitly
(
−
1
)
(
1
)
2
(
1
)
2
=
−
1
{\displaystyle {\frac {(-1)(1)^{2}}{(1)^{2}}}=-1}
. Thus they concluded that the pion is a pseudoscalar
particle.
=== Parity violation ===
Although parity is conserved in electromagnetism,
strong interactions and gravity, it turns
out to be violated in weak interactions. The
Standard Model incorporates parity violation
by expressing the weak interaction as a chiral
gauge interaction. Only the left-handed components
of particles and right-handed components of
antiparticles participate in weak interactions
in the Standard Model. This implies that parity
is not a symmetry of our universe, unless
a hidden mirror sector exists in which parity
is violated in the opposite way.
By the mid-20th century, it had been suggested
by several scientists that parity might not
be conserved (in different contexts), but
without solid evidence these suggestions were
not considered important. Then, in 1956, a
careful review and analysis by theoretical
physicists Tsung Dao Lee and Chen Ning Yang
went further, showing that while parity conservation
had been verified in decays by the strong
or electromagnetic interactions, it was untested
in the weak interaction. They proposed several
possible direct experimental tests. They were
mostly ignored, but Lee was able to convince
his Columbia colleague Chien-Shiung Wu to
try it. She needed special cryogenic facilities
and expertise, so the experiment was done
at the National Bureau of Standards.
In 1957 Wu, E. Ambler, R. W. Hayward, D. D.
Hoppes, and R. P. Hudson found a clear violation
of parity conservation in the beta decay of
cobalt-60. As the experiment was winding down,
with double-checking in progress, Wu informed
Lee and Yang of their positive results, and
saying the results need further examination,
she asked them not to publicize the results
first. However, Lee revealed the results to
his Columbia colleagues on 4 January 1957
at a "Friday Lunch" gathering of the Physics
Department of Columbia. Three of them, R.
L. Garwin, Leon Lederman, and R. Weinrich
modified an existing cyclotron experiment,
and they immediately verified the parity violation.
They delayed publication of their results
until after Wu's group was ready, and the
two papers appeared back to back in the same
physics journal.
After the fact, it was noted that an obscure
1928 experiment had in effect reported parity
violation in weak decays, but since the appropriate
concepts had not yet been developed, those
results had no impact. The discovery of parity
violation immediately explained the outstanding
τ–θ puzzle in the physics of kaons.
In 2010, it was reported that physicists working
with the Relativistic Heavy Ion Collider (RHIC)
had created a short-lived parity symmetry-breaking
bubble in quark-gluon plasmas. An experiment
conducted by several physicists including
Yale's Jack Sandweiss as part of the STAR
collaboration, suggested that parity may also
be violated in the strong interaction.
=== Intrinsic parity of hadrons ===
To every particle one can assign an intrinsic
parity as long as nature preserves parity.
Although weak interactions do not, one can
still assign a parity to any hadron by examining
the strong interaction reaction that produces
it, or through decays not involving the weak
interaction, such as rho meson decay to pions.
== See also ==
Electroweak theory
Standard Model
Mirror matter
