Symmetries in quantum mechanics describe features
of spacetime and particles which are unchanged
under some transformation, in the context
of quantum mechanics, relativistic quantum
mechanics and quantum field theory, and with
applications in the mathematical formulation
of the standard model and condensed matter
physics. In general, symmetry in physics,
invariance, and conservation laws, are fundamentally
important constraints for formulating physical
theories and models. In practice, they are
powerful methods for solving problems and
predicting what can happen. While conservation
laws do not always give the answer to the
problem directly, they form the correct constraints
and the first steps to solving a multitude
of problems.
This article outlines the connection between
the classical form of continuous symmetries
as well as their quantum operators, and relates
them to the Lie groups, and relativistic transformations
in the Lorentz group and Poincaré group.
== Notation ==
The notational conventions used in this article
are as follows. Boldface indicates vectors,
four vectors, matrices, and vectorial operators,
while quantum states use bra–ket notation.
Wide hats are for operators, narrow hats are
for unit vectors (including their components
in tensor index notation). The summation convention
on the repeated tensor indices is used, unless
stated otherwise. The Minkowski metric signature
is (+−−−).
== Symmetry transformations on the wavefunction
in non-relativistic quantum mechanics ==
=== Continuous symmetries ===
Generally, the correspondence between continuous
symmetries and conservation laws is given
by Noether's theorem.
The form of the fundamental quantum operators,
for example energy as a partial time derivative
and momentum as a spatial gradient, becomes
clear when one considers the initial state,
then changes one parameter of it slightly.
This can be done for displacements (lengths),
durations (time), and angles (rotations).
Additionally, the invariance of certain quantities
can be seen by making such changes in lengths
and angles, which illustrates conservation
of these quantities.
In what follows, transformations on only one-particle
wavefunctions in the form:
Ω
^
ψ
(
r
,
t
)
=
ψ
(
r
′
,
t
′
)
{\displaystyle {\widehat {\Omega }}\psi (\mathbf
{r} ,t)=\psi (\mathbf {r} ',t')}
are considered, where
Ω
^
{\displaystyle {\widehat {\Omega }}}
denotes a unitary operator. Unitarity is generally
required for operators representing transformations
of space, time, and spin, since the norm of
a state (representing the total probability
of finding the particle somewhere with some
spin) must be invariant under these transformations.
The inverse is the Hermitian conjugate
Ω
^
−
1
=
Ω
^
†
{\displaystyle {\widehat {\Omega }}^{-1}={\widehat
{\Omega }}^{\dagger }}
. The results can be extended to many-particle
wavefunctions. Written in Dirac notation as
standard, the transformations on quantum state
vectors are:
Ω
^
|
r
(
t
)
⟩
=
|
r
′
(
t
′
)
⟩
{\displaystyle {\widehat {\Omega }}\left|\mathbf
{r} (t)\right\rangle =\left|\mathbf {r} '(t')\right\rangle
}
Now, the action of
Ω
^
{\displaystyle {\widehat {\Omega }}}
changes ψ(r, t) to ψ(r′, t′), so the
inverse
Ω
^
−
1
=
Ω
^
†
{\displaystyle {\widehat {\Omega }}^{-1}={\widehat
{\Omega }}^{\dagger }}
changes ψ(r′, t′) back to ψ(r, t), so
an operator
A
^
{\displaystyle {\widehat {A}}}
invariant under
Ω
^
{\displaystyle {\widehat {\Omega }}}
satisfies:
A
^
ψ
=
Ω
^
†
A
^
Ω
^
ψ
⇒
Ω
^
A
^
ψ
=
A
^
Ω
^
ψ
{\displaystyle {\widehat {A}}\psi ={\widehat
{\Omega }}^{\dagger }{\widehat {A}}{\widehat
{\Omega }}\psi \quad \Rightarrow \quad {\widehat
{\Omega }}{\widehat {A}}\psi ={\widehat {A}}{\widehat
{\Omega }}\psi }
and thus:
[
Ω
^
,
A
^
]
ψ
=
0
{\displaystyle [{\widehat {\Omega }},{\widehat
{A}}]\psi =0}
for any state ψ. Quantum operators representing
observables are also required to be Hermitian
so that their eigenvalues are real numbers,
i.e. the operator equals its Hermitian conjugate,
A
^
=
A
^
†
{\displaystyle {\widehat {A}}={\widehat {A}}^{\dagger
}}
.
=== Overview of Lie group theory ===
Following are the key points of group theory
relevant to quantum theory, examples are given
throughout the article. For an alternative
approach using matrix groups, see the books
of HallLet G be a Lie group, which is a group
that locally is parameterized by a finite
number N of real continuously varying parameters
ξ1, ξ2, ... ξN. In more mathematical language,
this means that G is a smooth manifold that
is also a group, for which the group operations
are smooth.
the dimension of the group, N, is the number
of parameters it has.
the group elements, g, in G are functions
of the parameters:
g
=
G
(
ξ
1
,
ξ
2
,
⋯
)
{\displaystyle g=G(\xi _{1},\xi _{2},\cdots
)}
and all parameters set to zero returns the
identity element of the group:
I
=
G
(
0
,
0
⋯
)
{\displaystyle I=G(0,0\cdots )}
Group elements are often matrices which act
on vectors, or transformations acting on functions.The
generators of the group are the partial derivatives
of the group elements with respect to the
group parameters with the result evaluated
when the parameter is set to zero:
X
j
=
∂
g
∂
ξ
j
|
ξ
j
=
0
{\displaystyle X_{j}=\left.{\frac {\partial
g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}}
In the language of manifolds, the generators
are the elements of the tangent space to G
at the identity. The generators are also known
as infinitesimal group elements or as the
elements of the Lie algebra of G. (See the
discussion below of the commutator.)
One aspect of generators in theoretical physics
is they can be construed themselves as operators
corresponding to symmetries, which may be
written as matrices, or as differential operators.
In quantum theory, for unitary representations
of the group, the generators require a factor
of i:
X
j
=
i
∂
g
∂
ξ
j
|
ξ
j
=
0
{\displaystyle X_{j}=i\left.{\frac {\partial
g}{\partial \xi _{j}}}\right|_{\xi _{j}=0}}
The generators of the group form a vector
space, which means linear combinations of
generators also form a generator.The generators
(whether matrices or differential operators)
satisfy the commutation relations:
[
X
a
,
X
b
]
=
i
f
a
b
c
X
c
{\displaystyle \left[X_{a},X_{b}\right]=if_{abc}X_{c}}
where fabc are the (basis dependent) structure
constants of the group. This makes, together
with the vector space property, the set of
all generators of a group a Lie algebra. Due
to the antisymmetry of the bracket, the structure
constants of the group are antisymmetric in
the first two indices.The representations
of the group then describe the ways that the
group G (or its Lie algebra) can act on a
vector space. (The vector space might be,
for example, the space of eigenvectors for
a Hamiltonian having G as its symmetry group.)
We denote the representations using a capital
D. One can then differentiate D to obtain
a representation of the Lie algebra, often
also denoted by D. These two representations
are related as follows:
D
[
g
(
ξ
j
)
]
≡
D
(
ξ
j
)
=
e
i
ξ
j
D
(
X
j
)
{\displaystyle D[g(\xi _{j})]\equiv D(\xi
_{j})=e^{i\xi _{j}D(X_{j})}}
without summation on the repeated index j.
Representations are linear operators that
take in group elements and preserve the composition
rule:
D
(
ξ
a
)
D
(
ξ
b
)
=
D
(
ξ
a
ξ
b
)
.
{\displaystyle D(\xi _{a})D(\xi _{b})=D(\xi
_{a}\xi _{b}).}
A representation which cannot be decomposed
into a direct sum of other representations,
is called irreducible. It is conventional
to label irreducible representations by a
superscripted number n in brackets, as in
D(n), or if there is more than one number,
we write D(n, m, ... ).
There is an additional subtlety that arises
in quantum theory, where two vectors that
differ by multiplication by a scalar represent
the same physical state. Here, the pertinent
notion of representation is a projective representation,
one that only satisfies the composition law
up to a scalar. In the context of quantum
mechanical spin, such representations are
called spinorial.
=== Momentum and energy as generators of translation
and time evolution, and rotation ===
The space translation operator
T
^
(
Δ
r
)
{\displaystyle {\widehat {T}}(\Delta \mathbf
{r} )}
acts on a wavefunction to shift the space
coordinates by an infinitesimal displacement
Δr. The explicit expression
T
^
{\displaystyle {\widehat {T}}}
can be quickly determined by a Taylor expansion
of ψ(r + Δr, t) about r, then (keeping the
first order term and neglecting second and
higher order terms), replace the space derivatives
by the momentum operator
p
^
{\displaystyle {\widehat {\mathbf {p} }}}
. Similarly for the time translation operator
acting on the time parameter, the Taylor expansion
of ψ(r, t + Δt) is about t, and the time
derivative replaced by the energy operator
E
^
{\displaystyle {\widehat {E}}}
.
The exponential functions arise by definition
as those limits, due to Euler, and can be
understood physically and mathematically as
follows. A net translation can be composed
of many small translations, so to obtain the
translation operator for a finite increment,
replace Δr by Δr/N and Δt by Δt/N, where
N is a positive non-zero integer. Then as
N increases, the magnitude of Δr and Δt
become even smaller, while leaving the directions
unchanged. Acting the infinitesimal operators
on the wavefunction N times and taking the
limit as N tends to infinity gives the finite
operators.
Space and time translations commute, which
means the operators and generators commute.
For a time-independent Hamiltonian, energy
is conserved in time and quantum states are
stationary states: the eigenstates of the
Hamiltonian are the energy eigenvalues E:
U
^
(
t
)
=
exp
⁡
(
−
i
Δ
t
E
ℏ
)
{\displaystyle {\widehat {U}}(t)=\exp \left(-{\frac
{i\Delta tE}{\hbar }}\right)}
and all stationary states have the form
ψ
(
r
,
t
+
t
0
)
=
U
^
(
t
−
t
0
)
ψ
(
r
,
t
0
)
{\displaystyle \psi (\mathbf {r} ,t+t_{0})={\widehat
{U}}(t-t_{0})\psi (\mathbf {r} ,t_{0})}
where t0 is the initial time, usually set
to zero since there is no loss of continuity
when the initial time is set.
An alternative notation is
U
^
(
t
−
t
0
)
≡
U
(
t
,
t
0
)
{\displaystyle {\widehat {U}}(t-t_{0})\equiv
U(t,t_{0})}
.
=== Angular momentum as the generator of rotations
===
==== Orbital angular momentum ====
The rotation operator acts on a wavefunction
to rotate the spatial coordinates of a particle
by a constant angle Δθ:
R
(
Δ
θ
,
a
^
)
ψ
(
r
,
t
)
=
ψ
(
r
′
,
t
)
{\displaystyle {R}(\Delta \theta ,{\hat {\mathbf
{a} }})\psi (\mathbf {r} ,t)=\psi (\mathbf
{r} ',t)}
where r′ are the rotated coordinates about
an axis defined by a unit vector
a
^
=
(
a
1
,
a
2
,
a
3
)
{\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})}
through an angular increment Δθ, given by:
r
′
=
R
^
(
Δ
θ
,
a
^
)
r
.
{\displaystyle \mathbf {r} '={\widehat {R}}(\Delta
\theta ,{\hat {\mathbf {a} }})\mathbf {r}
\,.}
where
R
^
(
Δ
θ
,
a
^
)
{\displaystyle {\widehat {R}}(\Delta \theta
,{\hat {\mathbf {a} }})}
is a rotation matrix dependent on the axis
and angle. In group theoretic language, the
rotation matrices are group elements, and
the angles and axis
Δ
θ
a
^
=
Δ
θ
(
a
1
,
a
2
,
a
3
)
{\displaystyle \Delta \theta {\hat {\mathbf
{a} }}=\Delta \theta (a_{1},a_{2},a_{3})}
are the parameters, of the three-dimensional
special orthogonal group, SO(3). The rotation
matrices about the standard Cartesian basis
vector
e
^
x
,
e
^
y
,
e
^
z
{\displaystyle {\hat {\mathbf {e} }}_{x},{\hat
{\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z}}
through angle Δθ, and the corresponding
generators of rotations J = (Jx, Jy, Jz),
are:
More generally for rotations about an axis
defined by
a
^
{\displaystyle {\hat {\mathbf {a} }}}
, the rotation matrix elements are:
[
R
^
(
θ
,
a
^
)
]
i
j
=
(
δ
i
j
−
a
i
a
j
)
cos
⁡
θ
−
ε
i
j
k
a
k
sin
⁡
θ
+
a
i
a
j
{\displaystyle [{\widehat {R}}(\theta ,{\hat
{\mathbf {a} }})]_{ij}=(\delta _{ij}-a_{i}a_{j})\cos
\theta -\varepsilon _{ijk}a_{k}\sin \theta
+a_{i}a_{j}}
where δij is the Kronecker delta, and εijk
is the Levi-Civita symbol.
It is not as obvious how to determine the
rotational operator compared to space and
time translations. We may consider a special
case (rotations about the x, y, or z-axis)
then infer the general result, or use the
general rotation matrix directly and tensor
index notation with δij and εijk. To derive
the infinitesimal rotation operator, which
corresponds to small Δθ, we use the small
angle approximations sin(Δθ) ≈ Δθ and
cos(Δθ) ≈ 1, then Taylor expand about
r or ri, keep the first order term, and substitute
the angular momentum operator components.
The z-component of angular momentum can be
replaced by the component along the axis defined
by
a
^
{\displaystyle {\hat {\mathbf {a} }}}
, using the dot product
a
^
⋅
L
^
{\displaystyle {\hat {\mathbf {a} }}\cdot
{\widehat {\mathbf {L} }}}
.
Again, a finite rotation can be made from
lots of small rotations, replacing Δθ by
Δθ/N and taking the limit as N tends to
infinity gives the rotation operator for a
finite rotation.
Rotations about the same axis do commute,
for example a rotation through angles θ1
and θ2 about axis i can be written
R
(
θ
1
+
θ
2
,
e
i
)
=
R
(
θ
1
e
i
)
R
(
θ
2
e
i
)
,
[
R
(
θ
1
e
i
)
,
R
(
θ
2
e
i
)
]
=
0
.
{\displaystyle R(\theta _{1}+\theta _{2},\mathbf
{e} _{i})=R(\theta _{1}\mathbf {e} _{i})R(\theta
_{2}\mathbf {e} _{i})\,,\quad [R(\theta _{1}\mathbf
{e} _{i}),R(\theta _{2}\mathbf {e} _{i})]=0\,.}
However, rotations about different axes do
not commute. The general commutation rules
are summarized by
[
L
i
,
L
j
]
=
i
ℏ
ε
i
j
k
L
k
.
{\displaystyle [L_{i},L_{j}]=i\hbar \varepsilon
_{ijk}L_{k}.}
In this sense, orbital angular momentum has
the common sense properties of rotations.
Each of the above commutators can be easily
demonstrated by holding an everyday object
and rotating it through the same angle about
any two different axes in both possible orderings;
the final configurations are different.
In quantum mechanics, there is another form
of rotation which mathematically appears similar
to the orbital case, but has different properties,
described next.
==== Spin angular momentum ====
All previous quantities have classical definitions.
Spin is a quantity possessed by particles
in quantum mechanics without any classical
analogue, having the units of angular momentum.
The spin vector operator is denoted
S
^
=
(
S
x
^
,
S
y
^
,
S
z
^
)
{\displaystyle {\widehat {\mathbf {S} }}=({\widehat
{S_{x}}},{\widehat {S_{y}}},{\widehat {S_{z}}})}
. The eigenvalues of its components are the
possible outcomes (in units of
ℏ
{\displaystyle \hbar }
) of a measurement of the spin projected onto
one of the basis directions.
Rotations (of ordinary space) about an axis
a
^
{\displaystyle {\hat {\mathbf {a} }}}
through angle θ about the unit vector
a
^
{\displaystyle {\hat {a}}}
in space acting on a multicomponent wave function
(spinor) at a point in space is represented
by:
However, unlike orbital angular momentum in
which the z-projection quantum number ℓ
can only take positive or negative integer
values (including zero), the z-projection
spin quantum number s can take all positive
and negative half-integer values. There are
rotational matrices for each spin quantum
number.
Evaluating the exponential for a given z-projection
spin quantum number s gives a (2s + 1)-dimensional
spin matrix. This can be used to define a
spinor as a column vector of 2s + 1 components
which transforms to a rotated coordinate system
according to the spin matrix at a fixed point
in space.
For the simplest non-trivial case of s = 1/2,
the spin operator is given by
S
^
=
ℏ
2
σ
{\displaystyle {\widehat {\mathbf {S} }}={\frac
{\hbar }{2}}{\boldsymbol {\sigma }}}
where the Pauli matrices in the standard representation
are:
σ
1
=
σ
x
=
(
0
1
1
0
)
,
σ
2
=
σ
y
=
(
0
−
i
i
0
)
,
σ
3
=
σ
z
=
(
1
0
0
−
1
)
{\displaystyle \sigma _{1}=\sigma _{x}={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\,,\quad
\sigma _{2}=\sigma _{y}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\,,\quad
\sigma _{3}=\sigma _{z}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}
==== Total angular momentum ====
The total angular momentum operator is the
sum of the orbital and spin
J
^
=
L
^
+
S
^
{\displaystyle {\widehat {\mathbf {J} }}={\widehat
{\mathbf {L} }}+{\widehat {\mathbf {S} }}}
and is an important quantity for multi-particle
systems, especially in nuclear physics and
the 
quantum chemistry of multi-electron atoms
and molecules.
We have a similar rotation matrix:
J
^
(
θ
,
a
^
)
=
exp
⁡
(
−
i
ℏ
θ
a
^
⋅
J
^
)
{\displaystyle {\widehat {J}}(\theta ,{\hat
{\mathbf {a} }})=\exp \left(-{\frac {i}{\hbar
}}\theta {\hat {\mathbf {a} }}\cdot {\widehat
{\mathbf {J} }}\right)}
=== Conserved quantities in the Quantum Harmonic
Oscillator ===
The dynamical symmetry group of the n dimensional
quantum harmonic oscillator is the special
unitary group SU(n). As an example, the number
of infinitesimal generators of the corresponding
Lie algebras of SU(2) and SU(3) are three
and eight respectively. This leads to exactly
three and eight independent conserved quantities
(other than the Hamiltonian) in these systems.
The two dimensional quantum harmonic oscillator
has the expected conserved quantities of the
Hamiltonian and the angular momentum, but
has additional hidden conserved quantities
of energy level difference and another form
of angular momentum.
== Lorentz group in relativistic quantum mechanics
==
Following is an overview of the Lorentz group;
a treatment of boosts and rotations in spacetime.
Throughout this section, see (for example)
T. Ohlsson (2011) and E. Abers (2004).Lorentz
transformations can be parametrized by rapidity
φ for a boost in the direction of a three-dimensional
unit vector
n
^
=
(
n
1
,
n
2
,
n
3
)
{\displaystyle {\hat {\mathbf {n} }}=(n_{1},n_{2},n_{3})}
, and a rotation angle θ about a three-dimensional
unit vector
a
^
=
(
a
1
,
a
2
,
a
3
)
{\displaystyle {\hat {\mathbf {a} }}=(a_{1},a_{2},a_{3})}
defining an axis, so
φ
n
^
=
φ
(
n
1
,
n
2
,
n
3
)
{\displaystyle \varphi {\hat {\mathbf {n}
}}=\varphi (n_{1},n_{2},n_{3})}
and
θ
a
^
=
θ
(
a
1
,
a
2
,
a
3
)
{\displaystyle \theta {\hat {\mathbf {a} }}=\theta
(a_{1},a_{2},a_{3})}
are together six parameters of the Lorentz
group (three for rotations and three for boosts).
The Lorentz group is 6-dimensional.
=== Pure rotations in spacetime ===
The rotation matrices and rotation generators
considered above form the spacelike part of
a four-dimensional matrix, representing pure-rotation
Lorentz transformations. Three of the Lorentz
group elements
R
^
x
,
R
^
y
,
R
^
z
{\displaystyle {\widehat {R}}_{x},{\widehat
{R}}_{y},{\widehat {R}}_{z}}
and generators J = (J1, J2, J3) for pure rotations
are:
The rotation matrices act on any four vector
A = (A0, A1, A2, A3) and rotate the space-like
components according to
A
′
=
R
^
(
Δ
θ
,
n
^
)
A
{\displaystyle \mathbf {A} '={\widehat {R}}(\Delta
\theta ,{\hat {\mathbf {n} }})\mathbf {A}
}
leaving the time-like coordinate unchanged.
In matrix expressions, A is treated as a column
vector.
=== Pure boosts in spacetime ===
A boost with velocity ctanhφ in the x, y,
or z directions given by the standard Cartesian
basis vector
e
^
x
,
e
^
y
,
e
^
z
{\displaystyle {\hat {\mathbf {e} }}_{x},{\hat
{\mathbf {e} }}_{y},{\hat {\mathbf {e} }}_{z}}
, are the boost transformation matrices. These
matrices
B
^
x
,
B
^
y
,
B
^
z
{\displaystyle {\widehat {B}}_{x},{\widehat
{B}}_{y},{\widehat {B}}_{z}}
and the corresponding generators K = (K1,
K2, K3) are the remaining three group elements
and generators of the Lorentz group:
The boost matrices act on any four vector
A = (A0, A1, A2, A3) and mix the time-like
and the space-like components, according to:
A
′
=
B
^
(
φ
,
n
^
)
A
{\displaystyle \mathbf {A} '={\widehat {B}}(\varphi
,{\hat {\mathbf {n} }})\mathbf {A} }
The term "boost" refers to the relative velocity
between two frames, and is not to be conflated
with momentum as the generator of translations,
as explained below.
=== Combining boosts and rotations ===
Products of rotations give another rotation
(a frequent exemplification of a subgroup),
while products of boosts and boosts or of
rotations and boosts cannot be expressed as
pure boosts or pure rotations. In general,
any Lorentz transformation can be expressed
as a product of a pure rotation and a pure
boost. For more background see (for example)
B.R. Durney (2011) and H.L. Berk et al. and
references therein.
The boost and rotation generators have representations
denoted D(K) and D(J) respectively, the capital
D in this context indicates a group representation.
For the Lorentz group, the representations
D(K) and D(J) of the generators K and J fulfill
the following commutation rules.
In all commutators, the boost entities mixed
with those for rotations, although rotations
alone simply give another rotation. Exponentiating
the generators gives the boost and rotation
operators which combine into the general Lorentz
transformation, under which the spacetime
coordinates transform from one rest frame
to another boosted and/or rotating frame.
Likewise, exponentiating the representations
of the generators gives the representations
of the boost and rotation operators, under
which a particle's spinor field transforms.
In the literature, the boost generators K
and rotation generators J are sometimes combined
into one generator for Lorentz transformations
M, an antisymmetric four-dimensional matrix
with entries:
M
0
a
=
−
M
a
0
=
K
a
,
M
a
b
=
ε
a
b
c
J
c
.
{\displaystyle M^{0a}=-M^{a0}=K_{a}\,,\quad
M^{ab}=\varepsilon _{abc}J_{c}\,.}
and correspondingly, the boost and rotation
parameters are collected into another antisymmetric
four-dimensional matrix ω, with entries:
ω
0
a
=
−
ω
a
0
=
φ
n
a
,
ω
a
b
=
θ
ε
a
b
c
a
c
,
{\displaystyle \omega _{0a}=-\omega _{a0}=\varphi
n_{a}\,,\quad \omega _{ab}=\theta \varepsilon
_{abc}a_{c}\,,}
The general Lorentz transformation is then:
Λ
(
φ
,
n
^
,
θ
,
a
^
)
=
exp
⁡
(
−
i
2
ω
α
β
M
α
β
)
=
exp
⁡
[
−
i
2
(
φ
n
^
⋅
K
+
θ
a
^
⋅
J
)
]
{\displaystyle \Lambda (\varphi ,{\hat {\mathbf
{n} }},\theta ,{\hat {\mathbf {a} }})=\exp
\left(-{\frac {i}{2}}\omega _{\alpha \beta
}M^{\alpha \beta }\right)=\exp \left[-{\frac
{i}{2}}\left(\varphi {\hat {\mathbf {n} }}\cdot
\mathbf {K} +\theta {\hat {\mathbf {a} }}\cdot
\mathbf {J} \right)\right]}
with summation over repeated matrix indices
α and β. The Λ matrices act on any four
vector A = (A0, A1, A2, A3) and mix the time-like
and the space-like components, according to:
A
′
=
Λ
(
φ
,
n
^
,
θ
,
a
^
)
A
{\displaystyle \mathbf {A} '=\Lambda (\varphi
,{\hat {\mathbf {n} }},\theta ,{\hat {\mathbf
{a} }})\mathbf {A} }
=== Transformations of spinor wavefunctions
in relativistic quantum mechanics ===
In relativistic quantum mechanics, wavefunctions
are no longer single-component scalar fields,
but now 2(2s + 1) component spinor fields,
where s is the spin of the particle. The transformations
of these functions in spacetime are given
below.
Under a proper orthochronous Lorentz transformation
(r, t) → Λ(r, t) in Minkowski space, all
one-particle quantum states ψσ locally transform
under some representation D of the Lorentz
group:
ψ
σ
(
r
,
t
)
→
D
(
Λ
)
ψ
σ
(
Λ
−
1
(
r
,
t
)
)
{\displaystyle \psi _{\sigma }(\mathbf {r}
,t)\rightarrow D(\Lambda )\psi _{\sigma }(\Lambda
^{-1}(\mathbf {r} ,t))}
where D(Λ) is a finite-dimensional representation,
in other words a (2s + 1)×(2s + 1) dimensional
square matrix, and ψ is thought of as a column
vector containing components with the (2s
+ 1) allowed values of σ:
ψ
(
r
,
t
)
=
[
ψ
σ
=
s
(
r
,
t
)
ψ
σ
=
s
−
1
(
r
,
t
)
⋮
ψ
σ
=
−
s
+
1
(
r
,
t
)
ψ
σ
=
−
s
(
r
,
t
)
]
⇌
ψ
(
r
,
t
)
†
=
[
ψ
σ
=
s
(
r
,
t
)
⋆
ψ
σ
=
s
−
1
(
r
,
t
)
⋆
⋯
ψ
σ
=
−
s
+
1
(
r
,
t
)
⋆
ψ
σ
=
−
s
(
r
,
t
)
⋆
]
{\displaystyle \psi (\mathbf {r} ,t)={\begin{bmatrix}\psi
_{\sigma =s}(\mathbf {r} ,t)\\\psi _{\sigma
=s-1}(\mathbf {r} ,t)\\\vdots \\\psi _{\sigma
=-s+1}(\mathbf {r} ,t)\\\psi _{\sigma =-s}(\mathbf
{r} ,t)\end{bmatrix}}\quad \rightleftharpoons
\quad {\psi (\mathbf {r} ,t)}^{\dagger }={\begin{bmatrix}{\psi
_{\sigma =s}(\mathbf {r} ,t)}^{\star }&{\psi
_{\sigma =s-1}(\mathbf {r} ,t)}^{\star }&\cdots
&{\psi _{\sigma =-s+1}(\mathbf {r} ,t)}^{\star
}&{\psi _{\sigma =-s}(\mathbf {r} ,t)}^{\star
}\end{bmatrix}}}
=== Real irreducible representations and spin
===
The irreducible representations of D(K) and
D(J), in short "irreps", can be used to build
to spin representations of the Lorentz group.
Defining new operators:
A
=
J
+
i
K
2
,
B
=
J
−
i
K
2
,
{\displaystyle \mathbf {A} ={\frac {\mathbf
{J} +i\mathbf {K} }{2}}\,,\quad \mathbf {B}
={\frac {\mathbf {J} -i\mathbf {K} }{2}}\,,}
so A and B are simply complex conjugates of
each other, it follows they satisfy the symmetrically
formed commutators:
[
A
i
,
A
j
]
=
ε
i
j
k
A
k
,
[
B
i
,
B
j
]
=
ε
i
j
k
B
k
,
[
A
i
,
B
j
]
=
0
,
{\displaystyle \left[A_{i},A_{j}\right]=\varepsilon
_{ijk}A_{k}\,,\quad \left[B_{i},B_{j}\right]=\varepsilon
_{ijk}B_{k}\,,\quad \left[A_{i},B_{j}\right]=0\,,}
and these are essentially the commutators
the orbital and spin angular momentum operators
satisfy. Therefore, A and B form operator
algebras analogous to angular momentum; same
ladder operators, z-projections, etc., independently
of each other as each of their components
mutually commute. By the analogy to the spin
quantum number, we can introduce positive
integers or half integers, a, b, with corresponding
sets of values m = a, a − 1, ... −a +
1, −a and n = b, b − 1, ... −b + 1,
−b. The matrices satisfying the above commutation
relations are the same as for spins a and
b have components given by multiplying Kronecker
delta values with angular momentum matrix
elements:
(
A
x
)
m
′
n
′
,
m
n
=
δ
n
′
n
(
J
x
(
m
)
)
m
′
m
(
B
x
)
m
′
n
′
,
m
n
=
δ
m
′
m
(
J
x
(
n
)
)
n
′
n
{\displaystyle \left(A_{x}\right)_{m'n',mn}=\delta
_{n'n}\left(J_{x}^{(m)}\right)_{m'm}\,\quad
\left(B_{x}\right)_{m'n',mn}=\delta _{m'm}\left(J_{x}^{(n)}\right)_{n'n}}
(
A
y
)
m
′
n
′
,
m
n
=
δ
n
′
n
(
J
y
(
m
)
)
m
′
m
(
B
y
)
m
′
n
′
,
m
n
=
δ
m
′
m
(
J
y
(
n
)
)
n
′
n
{\displaystyle \left(A_{y}\right)_{m'n',mn}=\delta
_{n'n}\left(J_{y}^{(m)}\right)_{m'm}\,\quad
\left(B_{y}\right)_{m'n',mn}=\delta _{m'm}\left(J_{y}^{(n)}\right)_{n'n}}
(
A
z
)
m
′
n
′
,
m
n
=
δ
n
′
n
(
J
z
(
m
)
)
m
′
m
(
B
z
)
m
′
n
′
,
m
n
=
δ
m
′
m
(
J
z
(
n
)
)
n
′
n
{\displaystyle \left(A_{z}\right)_{m'n',mn}=\delta
_{n'n}\left(J_{z}^{(m)}\right)_{m'm}\,\quad
\left(B_{z}\right)_{m'n',mn}=\delta _{m'm}\left(J_{z}^{(n)}\right)_{n'n}}
where in each case the row number m′n′
and column number mn are separated by a comma,
and in turn:
(
J
z
(
m
)
)
m
′
m
=
m
δ
m
′
m
(
J
x
(
m
)
±
i
J
y
(
m
)
)
m
′
m
=
m
δ
a
′
,
a
±
1
(
a
∓
m
)
(
a
±
m
+
1
)
{\displaystyle \left(J_{z}^{(m)}\right)_{m'm}=m\delta
_{m'm}\,\quad \left(J_{x}^{(m)}\pm iJ_{y}^{(m)}\right)_{m'm}=m\delta
_{a',a\pm 1}{\sqrt {(a\mp m)(a\pm m+1)}}}
and similarly for J(n). The three J(m) matrices
are each (2m + 1)×(2m + 1) square matrices,
and the three J(n) are each (2n + 1)×(2n
+ 1) square matrices. The integers or half-integers
m and n numerate all the irreducible representations
by, in equivalent notations used by authors:
D(m, n) ≡ (m, n) ≡ D(m) ⊗ D(n), which
are each [(2m + 1)(2n + 1)]×[(2m + 1)(2n
+ 1)] square matrices.
Applying this to particles with spin s;
left-handed (2s + 1)-component spinors transform
under the real irreps D(s, 0),
right-handed (2s + 1)-component spinors transform
under the real irreps D(0, s),
taking direct sums symbolized by ⊕ (see
direct sum of matrices for the simpler matrix
concept), one obtains the representations
under which 2(2s + 1)-component spinors transform:
D(m, n) ⊕ D(n, m) where m + n = s. These
are also real irreps, but as shown above,
they split into complex conjugates.In these
cases the D refers to any of D(J), D(K), or
a full Lorentz transformation D(Λ).
=== Relativistic wave equations ===
In the context of the Dirac equation and Weyl
equation, the Weyl spinors satisfying the
Weyl equation transform under the simplest
irreducible spin representations of the Lorentz
group, since the spin quantum number in this
case is the smallest non-zero number allowed:
1/2. The 2-component left-handed Weyl spinor
transforms under D(1/2, 0) and the 2-component
right-handed Weyl spinor transforms under
D(0, 1/2). Dirac spinors satisfying the Dirac
equation transform under the representation
D(1/2, 0) ⊕ D(0, 1/2), the direct sum of
the irreps for the Weyl spinors.
== The Poincaré group in relativistic quantum
mechanics and field theory ==
Space translations, time translations, rotations,
and boosts, all taken together, constitute
the Poincaré group. The group elements are
the three rotation matrices and three boost
matrices (as in the Lorentz group), and one
for time translations and three for space
translations in spacetime. There is a generator
for each. Therefore, the Poincaré group is
10-dimensional.
In special relativity, space and time can
be collected into a four-position vector X
= (ct, −r), and in parallel so can energy
and momentum which combine into a four-momentum
vector P = (E/c, −p). With relativistic
quantum mechanics in mind, the time duration
and spatial displacement parameters (four
in total, one for time and three for space)
combine into a spacetime displacement ΔX
= (cΔt, −Δr), and the energy and momentum
operators are inserted in the four-momentum
to obtain a four-momentum operator,
P
^
=
(
E
^
c
,
−
p
^
)
=
i
ℏ
(
1
c
∂
∂
t
,
∇
)
,
{\displaystyle {\widehat {\mathbf {P} }}=\left({\frac
{\widehat {E}}{c}},-{\widehat {\mathbf {p}
}}\right)=i\hbar \left({\frac {1}{c}}{\frac
{\partial }{\partial t}},\nabla \right)\,,}
which are the generators of spacetime translations
(four in total, one time and three space):
X
^
(
Δ
X
)
=
exp
⁡
(
−
i
ℏ
Δ
X
⋅
P
^
)
=
exp
⁡
[
−
i
ℏ
(
Δ
t
E
^
+
Δ
r
⋅
p
^
)
]
.
{\displaystyle {\widehat {X}}(\Delta \mathbf
{X} )=\exp \left(-{\frac {i}{\hbar }}\Delta
\mathbf {X} \cdot {\widehat {\mathbf {P} }}\right)=\exp
\left[-{\frac {i}{\hbar }}\left(\Delta t{\widehat
{E}}+\Delta \mathbf {r} \cdot {\widehat {\mathbf
{p} }}\right)\right]\,.}
There are commutation relations between the
components four-momentum P (generators of
spacetime translations), and angular momentum
M (generators of Lorentz transformations),
that define the Poincaré algebra:
[
P
μ
,
P
ν
]
=
0
{\displaystyle [P_{\mu },P_{\nu }]=0\,}
1
i
[
M
μ
ν
,
P
ρ
]
=
η
μ
ρ
P
ν
−
η
ν
ρ
P
μ
{\displaystyle {\frac {1}{i}}[M_{\mu \nu },P_{\rho
}]=\eta _{\mu \rho }P_{\nu }-\eta _{\nu \rho
}P_{\mu }\,}
1
i
[
M
μ
ν
,
M
ρ
σ
]
=
η
μ
ρ
M
ν
σ
−
η
μ
σ
M
ν
ρ
−
η
ν
ρ
M
μ
σ
+
η
ν
σ
M
μ
ρ
{\displaystyle {\frac {1}{i}}[M_{\mu \nu },M_{\rho
\sigma }]=\eta _{\mu \rho }M_{\nu \sigma }-\eta
_{\mu \sigma }M_{\nu \rho }-\eta _{\nu \rho
}M_{\mu \sigma }+\eta _{\nu \sigma }M_{\mu
\rho }\,}
where η is the Minkowski metric tensor. (It
is common to drop any hats for the four-momentum
operators in the commutation relations). These
equations are an expression of the fundamental
properties of space and time as far as they
are known today. They have a classical counterpart
where the commutators are replaced by Poisson
brackets.
To describe spin in relativistic quantum mechanics,
the Pauli–Lubanski pseudovector
W
μ
=
1
2
ε
μ
ν
ρ
σ
J
ν
ρ
P
σ
,
{\displaystyle W_{\mu }={\frac {1}{2}}\varepsilon
_{\mu \nu \rho \sigma }J^{\nu \rho }P^{\sigma
},}
a Casimir operator, is the constant spin contribution
to the total angular momentum, and there are
commutation relations between P and W and
between M and W:
[
P
μ
,
W
ν
]
=
0
,
{\displaystyle \left[P^{\mu },W^{\nu }\right]=0\,,}
[
J
μ
ν
,
W
ρ
]
=
i
(
η
ρ
ν
W
μ
−
η
ρ
μ
W
ν
)
,
{\displaystyle \left[J^{\mu \nu },W^{\rho
}\right]=i\left(\eta ^{\rho \nu }W^{\mu }-\eta
^{\rho \mu }W^{\nu }\right)\,,}
[
W
μ
,
W
ν
]
=
−
i
ϵ
μ
ν
ρ
σ
W
ρ
P
σ
.
{\displaystyle \left[W_{\mu },W_{\nu }\right]=-i\epsilon
_{\mu \nu \rho \sigma }W^{\rho }P^{\sigma
}\,.}
Invariants constructed from W, instances of
Casimir invariants can be used to classify
irreducible representations of the Lorentz
group.
== Symmetries in quantum field theory and
particle physics ==
=== Unitary groups in quantum field theory
===
Group theory is an abstract way of mathematically
analyzing symmetries. Unitary operators are
paramount to quantum theory, so unitary groups
are important in particle physics. The group
of N dimensional unitary square matrices is
denoted U(N). Unitary operators preserve inner
products which means probabilities are also
preserved, so the quantum mechanics of the
system is invariant under unitary transformations.
Let
U
^
{\displaystyle {\widehat {U}}}
be a unitary operator, so the inverse is the
Hermitian adjoint
U
^
=
U
^
†
{\displaystyle {\widehat {U}}={\widehat {U}}^{\dagger
}}
, which commutes with the Hamiltonian:
[
U
^
,
H
^
]
=
0
{\displaystyle \left[{\widehat {U}},{\widehat
{H}}\right]=0}
then the observable corresponding to the operator
U
^
{\displaystyle {\widehat {U}}}
is conserved, and the Hamiltonian is invariant
under the transformation
U
^
{\displaystyle {\widehat {U}}}
.
Since the predictions of quantum mechanics
should be invariant under the action of a
group, physicists look for unitary transformations
to represent the group.
Important subgroups of each U(N) are those
unitary matrices which have unit determinant
(or are "unimodular"): these are called the
special unitary groups and are denoted SU(N).
==== U(1) ====
The simplest unitary group is U(1), which
is just the complex numbers of modulus 1.
This one-dimensional matrix entry is of the
form:
U
=
e
−
i
θ
{\displaystyle U=e^{-i\theta }}
in which θ is the parameter of the group,
and the group is Abelian since one-dimensional
matrices always commute under matrix multiplication.
Lagrangians in quantum field theory for complex
scalar fields are often invariant under U(1)
transformations. If there is a quantum number
a associated with the U(1) symmetry, for example
baryon and the three lepton numbers in electromagnetic
interactions, we have:
U
=
e
−
i
a
θ
{\displaystyle U=e^{-ia\theta }}
==== U(2) and SU(2) ====
The general form of an element of a U(2) element
is parametrized by two complex numbers a and
b:
U
=
(
a
b
−
b
⋆
a
⋆
)
{\displaystyle U={\begin{pmatrix}a&b\\-b^{\star
}&a^{\star }\\\end{pmatrix}}}
and for SU(2), the determinant is restricted
to 1:
det
(
U
)
=
a
a
⋆
+
b
b
⋆
=
|
a
|
2
+
|
b
|
2
=
1
{\displaystyle \det(U)=aa^{\star }+bb^{\star
}={|a|}^{2}+{|b|}^{2}=1}
In group theoretic language, the Pauli matrices
are the generators of the special unitary
group in two dimensions, denoted SU(2). Their
commutation relation is the same as for orbital
angular momentum, aside from a factor of 2:
[
σ
a
,
σ
b
]
=
2
i
ℏ
ε
a
b
c
σ
c
{\displaystyle [\sigma _{a},\sigma _{b}]=2i\hbar
\varepsilon _{abc}\sigma _{c}}
A group element of SU(2) can be written:
U
(
θ
,
e
^
j
)
=
e
i
θ
σ
j
/
2
{\displaystyle U(\theta ,{\hat {\mathbf {e}
}}_{j})=e^{i\theta \sigma _{j}/2}}
where σj is a Pauli matrix, and the group
parameters are the angles turned through about
an axis.
The two-dimensional isotropic quantum harmonic
oscillator has symmetry group SU(2), while
the symmetry algebra of the rational anisotropic
QHO is a nonlinear extension of u(2).
==== U(3) and SU(3) ====
The eight Gell-Mann matrices λn (see article
for them and the structure constants) are
important for quantum chromodynamics. They
originally arose in the theory SU(3) of flavor
which is still of practical importance in
nuclear physics. They are the generators for
the SU(3) group, so an element of SU(3) can
be written analogously to an element of SU(2):
U
(
θ
,
e
^
j
)
=
exp
⁡
(
−
i
2
∑
n
=
1
8
θ
n
λ
n
)
{\displaystyle U(\theta ,{\hat {\mathbf {e}
}}_{j})=\exp \left(-{\frac {i}{2}}\sum _{n=1}^{8}\theta
_{n}\lambda _{n}\right)}
where θn are eight independent parameters.
The λn matrices satisfy the commutator:
[
λ
a
,
λ
b
]
=
2
i
f
a
b
c
λ
c
{\displaystyle \left[\lambda _{a},\lambda
_{b}\right]=2if_{abc}\lambda _{c}}
where the indices a, b, c take the values
1, 2, 3... 8. The structure constants fabc
are totally antisymmetric in all indices analogous
to those of SU(2). In the standard colour
charge basis (r for red, g for green, b for
blue):
|
r
⟩
=
(
1
0
0
)
,
|
g
⟩
=
(
0
1
0
)
,
|
b
⟩
=
(
0
0
1
)
{\displaystyle |r\rangle ={\begin{pmatrix}1\\0\\0\end{pmatrix}}\,,\quad
|g\rangle ={\begin{pmatrix}0\\1\\0\end{pmatrix}}\,,\quad
|b\rangle ={\begin{pmatrix}0\\0\\1\end{pmatrix}}}
the colour states are eigenstates of the λ3
and λ8 matrices, while the other matrices
mix colour states together.
The eight gluons states (8-dimensional column
vectors) are simultaneous eigenstates of the
adjoint representation of SU(3) , the 8-dimensional
representation acting on its own Lie algebra
su(3), for the λ3 and λ8 matrices. By forming
tensor products of representations (the standard
representation and its dual) and taking appropriate
quotients, protons and neutrons, and other
hadrons are eigenstates of various representations
of SU(3) of color. The representations of
SU(3) can be described by a "theorem of the
highest weight".
=== Matter and antimatter ===
In relativistic quantum mechanics, relativistic
wave equations predict a remarkable symmetry
of nature: that every particle has a corresponding
antiparticle. This is mathematically contained
in the spinor fields which are the solutions
of the relativistic wave equations.
Charge conjugation switches particles and
antiparticles. Physical laws and interactions
unchanged by this operation have C symmetry.
=== Discrete spacetime symmetries ===
Parity mirrors the orientation of the spatial
coordinates from left-handed to right-handed.
Informally, space is "reflected" into its
mirror image. Physical laws and interactions
unchanged by this operation have P symmetry.
Time reversal flips the time coordinate, which
amounts to time running from future to past.
A curious property of time, which space does
not have, is that it is unidirectional: particles
traveling forwards in time are equivalent
to antiparticles traveling back in time. Physical
laws and interactions unchanged by this operation
have T symmetry.
=== C, P, T symmetries ===
CPT theorem
CP violation
PT symmetry
Lorentz violation
=== Gauge theory ===
In quantum electrodynamics, the symmetry group
is U(1) and is abelian. In quantum chromodynamics,
the symmetry group is SU(3) and is non-abelian.
The electromagnetic interaction is mediated
by photons, which have no electric charge.
The electromagnetic tensor has an electromagnetic
four-potential field possessing gauge symmetry.
The strong (color) interaction is mediated
by gluons, which can have eight color charges.
There are eight gluon field strength tensors
with corresponding gluon four potentials field,
each possessing gauge symmetry.
=== The strong (color) interaction ===
==== Color charge ====
Analogous to the spin operator, there are
color charge operators in terms of the Gell-Mann
matrices λj:
F
^
j
=
1
2
λ
j
{\displaystyle {\hat {F}}_{j}={\frac {1}{2}}\lambda
_{j}}
and since color charge is a conserved charge,
all color charge operators must commute with
the Hamiltonian:
[
F
^
j
,
H
^
]
=
0
{\displaystyle \left[{\hat {F}}_{j},{\hat
{H}}\right]=0}
==== Isospin ====
Isospin is conserved in strong interactions.
=== The weak and electromagnetic interactions
===
==== Duality transformation ====
Magnetic monopoles can be theoretically realized,
although current observations and theory are
consistent with them existing or not existing.
Electric and magnetic charges can effectively
be "rotated into one another" by a duality
transformation.
==== Electroweak symmetry ====
Electroweak symmetry
Electroweak symmetry breaking
=== Supersymmetry ===
A Lie superalgebra is an algebra in which
(suitable) basis elements either have a commutation
relation or have an anticommutation relation.
Symmetries have been proposed to the effect
that all fermionic particles have bosonic
analogues, and vice versa. These symmetry
have theoretical appeal in that no extra assumptions
(such as existence of strings) barring symmetries
are made. In addition, by assuming supersymmetry,
a number puzzling issues can be resolved.
These symmetries, which are represented by
Lie superalgebras, have not been confirmed
experimentally. It is now believed that they
are broken symmetries, if they exist. But
it has been speculated that dark matter is
constitutes gravitinos, a spin 3/2 particle
with mass, its supersymmetric partner being
the graviton.
== Exchange symmetry ==
The concept of exchange symmetry is derived
from a fundamental postulate of quantum statistics,
which states that no observable physical quantity
should change after exchanging two identical
particles. It states that because all observables
are proportional to
|
ψ
|
2
{\displaystyle \left|\psi \right|^{2}}
for a system of identical particles, the wave
function
ψ
{\displaystyle \psi }
must either remain the same or change sign
upon such an exchange.
Because the exchange of two identical particles
is mathematically equivalent to the rotation
of each particle by 180 degrees (and so to
the rotation of one particle's frame by 360
degrees), the symmetric nature of the wave
function depends on the particle's spin after
the rotation operator is applied to it. Integer
spin particles do not change the sign of their
wave function upon a 360 degree rotation—therefore
the sign of the wave function of the entire
system does not change. Semi-integer spin
particles change the sign of their wave function
upon a 360 degree rotation (see more in spin–statistics
theorem).
Particles for which the wave function does
not change sign upon exchange are called bosons,
or particles with a symmetric wave function.
The particles for which the wave function
of the system changes sign are called fermions,
or particles with an antisymmetric wave function.
Fermions therefore obey different statistics
(called Fermi–Dirac statistics) than bosons
(which obey Bose–Einstein statistics). One
of the consequences of Fermi–Dirac statistics
is the exclusion principle for fermions—no
two identical fermions can share the same
quantum state (in other words, the wave function
of two identical fermions in the same state
is zero). This in turn results in degeneracy
pressure for fermions—the strong resistance
of fermions to compression into smaller volume.
This resistance gives rise to the “stiffness”
or “rigidity” of ordinary atomic matter
(as atoms contain electrons which are fermions).
== See also ==
Projective representation
Casimir operator
Pauli–Lubanski pseudovector
Symmetries in general relativity
Renormalization group
Center of mass (relativistic)
Representation of a Lie group
Representation theory of the Poincaré group
Representation theory of the Lorentz group
== Footnotes
