User:YohanN7/Representation theory of some important groups

Representation theory of the Euclidean group E(2)

The Euclidean group E(2) in two dimensions is the group of isometries of the Euclidean plane. It is also denoted ISO(2) provided reflections are excluded. The I stands for inhomogeneous, referring to the translational part, and SO stands for special orthogonal, referring to the rotational part. Its elements are called rigid motions. When reflections are included, the group is sometimes denoted E⁺(2) (but rarely IO(2)). The elements are then motions.

Notation

Generic vectors in the plane are written in boldface latin letters $\mathbf {x} ,\mathbf {y} ,\ldots$. Constant vectors in the plane use $\mathbf {a} ,\mathbf {b} ,\ldots$ or subscripted versions. In matrix notation these are taken as column vectors.

Group multiplication rule

A rigid motion can be written as

$\mathbf {x} '=g\mathbf {x} =R\mathbf {x} +\mathbf {b} ,\quad \mathbf {x} ',\mathbf {x} ,\mathbf {b} \in \mathbb {R} ^{2},R\in \mathrm {SO} (2),g\in \mathrm {E} (2),$

where the vector is first rotated in the plane and then a translation is added. The group has a standard faithful three-dimensional representation. The idea is to embed ℝ² as the affine plane z = 1 in ℝ³. Then x ∈ ℝ² is represented by (x^T, 1)^T ∈ ℝ³, and

$\left({\begin{matrix}\mathbf {x} '\\1\end{matrix}}\right)=\left({\begin{matrix}R&\mathbf {b} \\0&1\end{matrix}}\right)\left({\begin{matrix}\mathbf {x} \\1\end{matrix}}\right)=\left({\begin{matrix}x'_{1}\\x'_{2}\\1\end{matrix}}\right)=\left({\begin{matrix}\cos \theta &-\sin \theta &b_{1}\\\sin \theta &\cos \theta &b_{2}\\0&0&1\end{matrix}}\right)\left({\begin{matrix}x_{1}\\x_{2}\\1\end{matrix}}\right)=\left({\begin{matrix}R\mathbf {x} +\mathbf {b} \\1\end{matrix}}\right).$

(GMR1)

The representation by the three-dimensional matrix above for $(-\infty <b_{1},b_{2}<\infty ,0\leq \theta <2\pi )$ is faithful.

The group multiplication rule

$(\mathbf {b} _{3},R_{3})=(\mathbf {b} _{2},R_{2})\cdot (\mathbf {b} _{1},R_{1})=(R_{2}\mathbf {b} _{1}+\mathbf {b} _{2},R_{2}R_{1}){\text{ or }}(\mathbf {b} _{3},\theta _{3})=(\mathbf {b} _{2},\theta _{2})\cdot (\mathbf {b} _{1},\theta _{1})=(R(\theta _{2})\mathbf {b} _{1}+\mathbf {b} _{2},\theta _{1}+\theta _{2})$

(GMR2)

follows by inspection of (GMR1), and the inverse operation is then

$(\mathbf {b} ,R)^{-1}=(-R^{-1}\mathbf {b} ,R^{-1}){\text{ or }}(\mathbf {b} ,\theta )^{-1}=(-R(-\theta )\mathbf {b} ,-\theta ).$

(GMR2)

Lie algebra

The Lie algebra representation is found, using the single generator $J$ of $\mathrm {SO} (2)$ and using the series representation of the matrix exponential, from the parametric matrix form of $g\in \mathrm {E} (2)$ above. The Lie algebra representation in this basis is

${\mathfrak {e}}(2)=\mathrm {span} \left\{J=\left({\begin{matrix}0&-1&0\\1&0&0\\0&0&0\end{matrix}}\right),\quad P_{1}=\left({\begin{matrix}0&0&1\\0&0&0\\0&0&0\end{matrix}}\right),\quad P_{2}=\left({\begin{matrix}0&0&0\\0&0&1\\0&0&0\end{matrix}}\right)\right\}.$

(LA1)

Direct computation yields the commutation relations

${\begin{aligned}[][P_{1},P_{2}]&=0,\\{}[J,P_{k}]&=\varepsilon _{km}P_{m},\end{aligned}}$

(LA2)

where $\varepsilon _{km}$ is the two-dimensional Levi-Civita symbol with $\varepsilon _{12}=1$.

Subalgebras

Two subalgebras can be identified, that spanned by $J$, isomorphic to ${\mathfrak {so}}(2)$, and that spanned by $P_{1}$ and $P_{2}$, here denoted ${\mathfrak {t}}(2)\approx \mathbb {R} ^{2}.$ Inspection of (LA2) shows that ${\mathfrak {t}}(2)$ is an ideal in ${\mathfrak {e}}(2).$ It follows that ${\mathfrak {e}}(2)$ semi-direct sum,

${\mathfrak {e}}(2)={\mathfrak {t}}(2)\oplus _{s}{\mathfrak {so}}(2).$

Correspondingly, $\mathrm {E} (2)$ is a semi-direct product,

$\mathrm {E} (2)=\mathrm {T} (2)\rtimes \mathrm {SO} (2),$

in which $\mathrm {T} ^{2}$ is a normal subgroup. The factor group is

$\mathrm {E} /\mathrm {T} \approx \mathrm {SO} (2).$

The adjoint action of $\mathrm {SO} (2)$ on $\mathrm {T} (2)$ is, using $\mathrm {Ad} _{e^{X}}=e^{\mathrm {ad} _{X}}$

$e^{\theta J}P_{k}e^{-\theta J}={R(\theta )^{m}}_{k}\mathrm {P} _{m}=\cos \theta P_{k}+\varepsilon _{km}\sin \theta P_{m},$

Proof

By the adjoint representation formula (proved here),

$e^{\theta J}P_{k}e^{-\theta J}=e^{\mathrm {ad} _{\theta J}}P_{k}.$

By (LA2),

${\begin{aligned}\mathrm {ad} _{\theta J}(P_{k})=\theta [J,P_{k}]&=\varepsilon _{km}\theta P_{m}\\\mathrm {ad} _{\theta J}^{2}(P_{i})=\theta ^{2}[J,[J,P_{i}]]=\theta ^{2}\varepsilon _{ij}[J,P_{j}]=\theta ^{2}\varepsilon _{ij}\varepsilon _{jk}P_{k}=-\delta _{ik}\theta ^{2}P_{k}&=-\theta ^{2}P_{i}.\end{aligned}}$

(LASP1)

Using the series expansion of the exponential map (Lie theory) and grouping terms

${\begin{aligned}e^{\mathrm {ad_{\theta J}} }P_{k}&=\sum _{n=0}^{\infty }{\frac {\mathrm {ad} _{\theta J}^{n}}{n!}}P_{k}\\&=\left(1+\mathrm {ad} _{\theta J}+{\frac {\mathrm {ad} _{\theta J}^{2}}{2!}}+{\frac {\mathrm {ad} _{\theta J}^{3}}{3!}}+\cdots \right)P_{k}\\&=\left(1+{\frac {\mathrm {ad} _{\theta J}^{2}}{2!}}+{\frac {\mathrm {ad} _{\theta J}^{4}}{4!}}+\cdots \right)P_{k}+\left(\mathrm {ad} _{\theta J}+{\frac {\mathrm {ad} _{\theta J}^{3}}{3!}}+{\frac {\mathrm {ad} _{\theta J}^{5}}{5!}}+\cdots \right)P_{k}.\end{aligned}}$

(LASP2)

Substituting (LASP1) in (LASP2) gives

${\begin{aligned}e^{\mathrm {ad_{\theta J}} }P_{k}&=\left(1+{\frac {-\theta ^{2}}{2!}}+{\frac {\theta ^{4}}{4!}}+\cdots \right)P_{k}+\left(\theta -{\frac {\theta ^{3}}{3!}}+{\frac {\theta ^{5}}{5!}}+\cdots \right)\varepsilon _{km}P_{m}.\\\end{aligned}}$

Recognizing the series expansion of the sine and the cosine, this is

$e^{\mathrm {ad_{\theta J}} }P_{k}=\cos \theta P_{k}+\varepsilon _{km}\sin \theta P_{m}.$

In matrix notation this becomes with

$R(\theta )=\left({\begin{matrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{matrix}}\right).$

in component notation of matrices

$e^{\mathrm {ad_{\theta J}} }P_{k}={R(\theta )^{m}}_{k}P_{m},$

and in pure matrix form, this is

$\mathbf {P} '^{\mathrm {T} }=\mathbf {P} ^{\mathrm {T} }R(\theta ).$

The effect on $\mathbf {b} \cdot \mathbf {P} =b^{k}P_{k}$ is seen to be

$e^{\mathrm {ad_{\theta J}} }b^{k}P_{k}={R(\theta )^{m}}_{k}b^{k}P_{m}=(R(\theta )\mathbf {b} )^{m}P_{m}=R(\theta )\mathbf {b} \cdot \mathbf {P} .$

---

leading to

$e^{\theta J}\mathbf {b} \cdot \mathbf {P} e^{-\theta J}=\mathbf {b} '\cdot \mathbf {P} ,\quad \mathbf {b} '=R(\theta )\mathbf {b} ,$

and hence

$e^{\theta J}e^{\mathbf {b} \cdot \mathbf {P} }e^{-\theta J}=e^{R(\theta )\mathbf {b} \cdot \mathbf {P} }.$

Casimir operator

The operator $P^{2}=P_{1}^{2}+P_{2}^{2}$ commutes with all Lie algebra elements since

$[J,P_{1}^{2}+P_{2}^{2}]=P_{1}[J,P_{1}]+[J,P_{1}]P_{1}+P_{2}[J,P_{2}]+[J,P_{2}]P_{2}=0,$

where (LA2) was used in the last step.

When unitary representations are assumed, The $P_{k}$ will be anti-Hermitian, meaning $P_{k}^{\dagger }=-P_{k}$, and hence $P^{2}$ will be positive-semidefinite. Its eigenvalues serve to partly classify the unitary representations.

Representation theory from the method of induced representations

• For each $m\in \mathbb {Z}$ there is a one-dimensional unitary representation of the full $\mathrm {E} (2).$ It is labeled by $(0,m)$, where the first coordinate refers to the eigenvalue of the Casimir operator $P^{2}$, and the second coordinate is a further label referring to the eigenvalue of the Casimir operator $J$ of the little group $\mathrm {SO} (2)$. The action of the Lie algebra is given by

${\begin{aligned}\pi _{m}(P_{1})\left|m\right\rangle &=0\left|m\right\rangle =\emptyset ,\\\pi _{m}(P_{2})\left|m\right\rangle &=0\left|m\right\rangle =\emptyset ,\\\pi _{m}(J)|m\rangle &=m|m\rangle .\end{aligned}}$

At the group level,

${\begin{aligned}e^{-i\mathbf {a} \cdot \mathbf {\pi } _{m}(P)}|m\rangle &=|m\rangle ,\\e^{-i\theta \pi _{m}(J)}|m\rangle &=e^{-im\theta }|m\rangle ,\end{aligned}}$

is obtained.

• For each $p>0\in \mathbb {Z}$ there is an infinite-dimensional unitary representation of the full $\mathrm {E} (2).$ It is labeled by $p$, the square root of eigenvalue of the Casimir operator. The action of the Lie algebra is given by

${\begin{aligned}P_{1}\left|p_{1},p_{2},\ldots \right\rangle &=\left|p_{1},p_{2},\ldots \right\rangle p_{1},\\P_{2}\left|p_{1},p_{2},\ldots \right\rangle &=\left|p_{1},p_{2},\ldots \right\rangle p_{2}\\J\left|\mathbf {p} \right\rangle &=\left|\mathbf {p} J\right\rangle .\end{aligned}}$

At the group level,

${\begin{aligned}e^{-i\mathbf {a} \cdot \mathbf {P} }\left|p_{1},p_{2},\ldots \right\rangle &=\left|p_{1},p_{2},\ldots \right\rangle e^{-i\mathbf {a} \cdot \mathbf {p} },\\R(\theta )\left|\mathbf {p} \right\rangle &=\left|\mathbf {p} R(\theta )\right\rangle .\end{aligned}}$

To derive these results, the standard representation on $\mathbb {R} ^{2}$ is examined for subgroups leaving invariant a vector $\mathbf {p} \in \mathbb {R} ^{2}$.

Little groups of Euclidean group E(2)

There are only two cases. Either $(p_{1},p_{2})=(0,0)$ in which case the little group is $\mathrm {SO} (2)$, or $(p_{1},p_{2})\neq (0,0)$ in which case the little group is the trivial group $1.$ The basis is chosen such that the the Hermitean representatives the commuting $P_{1},P_{2}$ are simultaneously diagonalized. This is called the linear momentum basis.

Nonzero vector: The one-element group

Here the labeling of states $\left|p_{1},p_{2},\ldots \right\rangle$ is introduced. By definition per above, the operators $\pi (P_{i})$ act by

${\begin{aligned}\pi (P_{1})\left|p_{1},p_{2},\ldots \right\rangle &=p_{1}\left|p_{1},p_{2},\ldots \right\rangle ,\\\pi (P_{2})\left|p_{1},p_{2},\ldots \right\rangle &=p_{2}\left|p_{1},p_{2},\ldots \right\rangle .\end{aligned}}$

The dots indicate possible further labels.

At the group level this is

$e^{-i\mathbf {a} \cdot \mathbf {P} }\left|p_{1},p_{2},\ldots \right\rangle =\left|p_{1},p_{2},\ldots \right\rangle e^{-i\mathbf {a} \cdot \mathbf {p} }.$

The Lie algebra ${\mathfrak {1}}$ of the little group $1$ is trivial, ${\mathfrak {1}}=\{\emptyset \},$ and $1$ has only one irreducible unitary representation, the trivial one.

To deduce the action of the full group $\mathrm {E} (2)$, the action of $\mathrm {SO} (2)$ is examined by examining the effect of $P_{k},k=1,2$ on rotated states. To facilitate notation, write $(p_{1},p_{2})$ as $\mathbf {p} .$

${\begin{aligned}P_{k}R(\theta )\left|\mathbf {p} ,\ldots \right\rangle &=\left[R(\theta )R(\theta )^{-1}\right]P_{k}R(\theta )\left|p_{1},p_{2},\ldots \right\rangle \\&=R(\theta )\left[R(\theta )^{-1}P_{k}R(\theta )\right]\left|p_{1},p_{2},\ldots \right\rangle \\&=R(\theta )\left[{R(-\theta )^{m}}_{k}P_{m}\right]\left|p_{1},p_{2},\ldots \right\rangle \\&=R(\theta )\left[{R(-\theta )^{1}}_{k}P_{1}+{R(-\theta )^{2}}_{k}P_{2}\right]\left|p_{1},p_{2},\ldots \right\rangle ,\\\end{aligned}}$

or

${\begin{aligned}P_{1}R(\theta )\left|\mathbf {p} ,\ldots \right\rangle &=R(\theta )\left[{R(-\theta )^{1}}_{1}P_{1}+{R(-\theta )^{2}}_{1}P_{2}\right]\left|p_{1},p_{2},\ldots \right\rangle =R(\theta )\left|p_{1},p_{2},\ldots \right\rangle \left[p_{1}\cos \theta -p_{2}\sin \theta \right],\\P_{2}R(\theta )\left|\mathbf {p} ,\ldots \right\rangle &=R(\theta )\left[{R(-\theta )^{1}}_{2}P_{1}+{R(-\theta )^{2}}_{2}P_{2}\right]\left|p_{1},p_{2},\ldots \right\rangle =R(\theta )\left|p_{1},p_{2},\ldots \right\rangle \left[p_{1}\sin \theta +p_{2}\cos \theta \right].\end{aligned}}$

On infinitesimal form, this is

${\begin{aligned}P_{1}R(\delta \theta )\left|\mathbf {p} ,\ldots \right\rangle &=R(\delta \theta )\left|p_{1},p_{2},\ldots \right\rangle \left[p_{1}\cos \delta \theta -p_{2}\sin \delta \theta \right]\approx R(\delta \theta )\left|p_{1},p_{2},\ldots \right\rangle \left[p_{1}-p_{2}\delta \theta \right],\\P_{2}R(\delta \theta )\left|\mathbf {p} ,\ldots \right\rangle &=R(\delta \theta )\left|p_{1},p_{2},\ldots \right\rangle \left[p_{1}\sin \delta \theta +p_{2}\cos \delta \theta \right]\approx R(\delta \theta )\left|p_{1},p_{2},\ldots \right\rangle \left[p_{1}\delta \theta +p_{2}\right].\end{aligned}}$

Since this is deduced from the postulated behavior of $P_{1}$ and $P_{2}$ on a single vector, and the result are eigenvalues different from the postulated ones for the single vector, the only reasonable conclusion is that $R(\theta )\left|\mathbf {p} \right\rangle$ is a new eigenvector of $P_{1}$ and $P_{2}$ orthogonal to $\mathbf {p}$. Evidently,

${\begin{aligned}R(\theta )\left|\mathbf {p} \right\rangle &=\left|R(\theta )\mathbf {p} \right\rangle \\J\left|\mathbf {p} \right\rangle &=\left|J\mathbf {p} \right\rangle .\end{aligned}}$

Since elements are orthogonal matrices, the norm of $\left|R(\theta )\mathbf {p} \right\rangle$ is the same as the norm of $\mathbf {p} .$ Thus the eigenvalue of the Casimir operator remains the same, and an infinite-dimensional unitary representation of $\mathrm {E} (2)$ is characterized by this eigenvalue.

Zero vector: Rotation group SO(2)

Here the labeling is $\left|0,0\,\ldots \right\rangle$ is introduced. The first two zeros refer to the eigenvalues of the $P_{i}$. They act by definition according to

${\begin{aligned}P_{1}\left|0,0,\ldots \right\rangle &=\left|0,0,\ldots \right\rangle 0=\emptyset ,\\P_{2}\left|0,0,\ldots \right\rangle &=\left|0,0,\ldots \right\rangle 0=\emptyset .\end{aligned}}$

It remains to work out how the little group acts. If the representation is to be irreducible, it must be one-dimensional, since only one-dimensional irreducible representations of $\mathbf {so} (2)$ exist. In these representations, labeled by $m\in \mathbb {Z}$ the generator $J$ of $\mathbf {so} (2)$ acts by

$J|0,0,...\rangle =|0,0,\cdots \rangle m.$

This suggests the labeling $\left|0,0,m\right\rangle ,m\in \mathbb {Z}$ for the basis vector.

At the group level, one obtains

${\begin{aligned}e^{-i\mathbf {a} \cdot \mathbf {P} }|0,0,m\rangle &=|0,0,m\rangle ,\\e^{-i\theta J}|0,0,m\rangle &=e^{-im\theta }|0,0,m\rangle ,\end{aligned}}$

As it happens, the actions of the abelian subgroup $\mathrm {T} (2)$ and of the little group $\mathrm {SO} (2)$ described so far exhausts the action of all of $\mathrm {E} (2)$.

References

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• Lomont, J. E.; Moses, H. E. (1962). "Simple Realizations of the Infinitesimal Generators of the Proper Orthochronous Inhomogeneous Lorentz Group for Mass Zero". J. Math. Phys. 3 (3). AIP Publishing LLC: 405–408. doi:10.1063/1.1724240. {{cite journal}}: Unknown parameter |subscription= ignored (|url-access= suggested) (help)

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