Field Automorphism is Normed Automorphism if and Only if It is Continuous

Automorphisms

the U ∈ G induce automorphisms that leave each F(O) globally fixed, and π(A(O))″ ⊆ F(O) is the set of fixed points under the action of G on F(O);

From: Philosophy of Physics , 2007

NON-RELATIVISTIC QUANTUM MECHANICS

Michael Dickson , in Philosophy of Physics, 2007

7.2.6 Unitary Operators

An automorphism of a vector space, V, is a map from V to itself that 'preserves the structure of V′, and in particular the linear, inner-product, and topological structures (the latter two if they exist in V). Let U be a (linear) operator on the Hilbert space $\mathcal{H}$ such that: U is invertible (hence U is 1-1); and U preserves inner products (i.e., for any $v,w\in \mathcal{H},〈Uw,Uv〉=〈w,v〉$ ). Such an operator is called 'unitary', and clearly implements an automorphism of $\mathcal{H}$ . In that case, of course U also preserves norms, i.e., ||Uv|| = ||v|| for all v.

It is readily shown that for any unitary operator, U, U* = U ⁻¹. Conversely, any invertible linear operator with that property is unitary. (If, instead, $〈Uw,Uv〉=〈w,v〉*$ and U (kv) = k*Uv, then U is anti-unitary.)

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ALGEBRAIC QUANTUM FIELD THEORY

Hans Halvorson , in Philosophy of Physics, 2007

NOTE 50

When considering the group Aut $\mathcal{A}$ of automorphisms of a C*-algebra, we take as our standard topology the strong topology on the set L ( $\mathcal{A}$ ) of bounded linear mappings on $\mathcal{A}$ (considered as a Banach space). That is, α _i converges to α just in case for each A ∈ $\mathcal{A}$ , α_i (A) converges to α(A) in the norm on $\mathcal{A}$ .

DEFINITION 51

We use the GNS representation theorem (Thm. 17) to transfer terminology about representations (Defn. 14) to terminology about states. So, e.g., we say that two states are disjoint if their GNS representations are disjoint.

A vacuum state should be at least translation invariant. Furthermore, the microcausality assumption on the net $\mathcal{A}$ entails that any two observables commute "in the limit" where one is translated out to spacelike infinity. That is, for any A, B ∈ $\mathcal{A}$ , and for any spacelike vector x,

$\underset{t\to \infty }{lim}\Vert \left[{\alpha }_{tx}\left(A\right),B\right]\Vert =0\cdot$

This in turn entails that G acts on $\mathcal{A}$ as a large group of automorphisms in the following sense:

If ω is a G-invariant state and ( $\mathcal{H}$ , π) is the GNS representation of $\mathcal{A}$ induced by ω, then for any A ∈ $\mathcal{A}$ ,

$\overline{\text{conv}}\{\pi \left({\alpha }_g\left(A\right)\right):g\in G,\}$

has nonempty intersection with π( $\mathcal{A}$ )′.

Here we use $\overline{\text{conv}}$ S to denote the weakly closed convex hull of S. (See [Størmer, 1970 for the relevant proofs.) Note however that we would also expect the same to be true in a non-relativistic setting, because we would expect observables associated with disjoint regions of space to commute. (We have not invoked the fact that any vector in Minkowski spacetime is the sum of two spacelike vectors.)

Thanks to extensive research on "C*-dynamical systems," much is known about G-invariant states when G acts as a large group of automorphisms of $\mathcal{A}$ . In particular, the set of G-invariant states is convex and closed (in the weak* topology), hence the set has extreme points, called extremal invariant states. (Obviously if a pure state of $\mathcal{A}$ is G-invariant, then it is extremal invariant.) Furthermore, we also have the following result concerning the disjointness of G-invariant states.

PROPOSITION 52

Let ω be a G-invariant state of $\mathcal{A}$ , let $\mathcal{H}$ be its GNS Hilbert space, and let Ω be the GNS vector. Then the following are equivalent:

1.

ω is clustering in the sense that

$\underset{t\to \infty }{lim}\omega \left({\alpha }_{tx}\left(A\right)B\right)=\omega \left(A\right)\omega \left(B\right)\cdot$

2.

ω is extremal invariant.

3.

If a G-invariant state ρ is quasiequivalent to ω, then ρ = ω. In other words, no other G-invariant state is quasiequivalent to ω.

4.

The ray spanned by ω is the unique (up to scalar multiples) G-invariant subspace of $\mathcal{H}$ .

Proof. See [Størmer, 1970]. For related details, see also [Emch, 1972, pp. 183, 287] and Emch, this volume, Section 3.

So, if a (vacuum) state is clustering, then no other translation invariant state is in its folium (i.e. the set of states that are quasiequivalent to that state). Similarly, if a state is extremal invariant (a fortiori if it is pure) then it is the unique translation invariant state in its folium.

NOTE 53

The existence of disjoint vacua is related to spontaneous symmetry breaking. See Section 10.7.

NOTE 54.

Prop. 52 plays a central role in the proof of "Haag's theorem" given in [Emch, 1972, p. 248]. In particular, the uniqueness of extremal G-invariant states is equated with the nonexistence of "vacuum polarization."

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ON SYMPLECTIC REDUCTION IN CLASSICAL MECHANICS

J. Butterfield , in Philosophy of Physics, 2007

6.1.1 Canonical actions and infinitesimal generators

Let G be a Lie group acting on a Poisson manifold M by a smooth left action ϕ: G × M → M; so that as usual we write ϕ _g : x ∈ M → ϕ _g (x):= g · x ∈ M. As in the definition of a Poisson map (eq. 283), we say the action is canonical if

(311) ${\text{Φ}}_g^{*}\left\{F_{\mathrm{-1}},F_2\right\}=\left\{{\text{Φ}}_g^{*}{F}_{\mathrm{-1}}{\text{Φ}}_g^{*}{F}_2\right\}$

for any F ₁, F ₂ ∈ F (M) and any g ∈ G. If M is symplectic with symplectic form ω, then the action is canonical iff it is symplectic, i.e. ϕ* _g ω = ω for all g ∈ G.

We will be especially interested in the infinitesimal version of this notion; and so with infinitesimal generators of actions. We recall from eq. 152 that the infinitesimal generator of the action corresponding to a Lie algebra element ξ ∈ g is the vector field ξ _M on M obtained by differentiating the action with respect to g at the identity in the direction ξ:

(312) ${\xi }_M\left(x\right)=\frac{d}{d\tau }\left[exp\left(\tau \xi \right)\cdot x\right]{|}_{\tau =0}\cdot$

So we differentiate eq. 311 with respect to g in the direction ξ, to give:

(313) ${\xi }_M\left(\left\{F_{\mathrm{-1}},F_2\right\}\right)=\left\{{\xi }_M\left(F_{\mathrm{-1}}\right),F_2\right\}+\left\{F_{\mathrm{-1}},{\xi }_M\left(F_2\right)\right\}\cdot$

Such a vector field ξ _M is called an infinitesimal Poisson automorphism.

Side-remark: We will shortly see that it is the universal quantification over g ∈ G in eq. 311, and correspondingly in eq. 313 and 315 below, that means our description of conserved quantities is no longer focussed on a single vector field; and in particular, that a momentum map representing a conserved quantity has components.

In the symplectic case, differentiating ϕ* _g ω = ω implies that the Lie derivative ${\mathcal{L}}_{\xi M}\omega$ of ω with respect to ξ vanishes: ${\mathcal{L}}_{\xi M}\omega =0$ . We saw in Section 2.1.3 that this is equivalent to ξ _M being locally Hamiltonian, i.e. there being a local scalar J: U ⊂ M → ℝ such that ξ _M = X_J . This was how Section 2.1.3 vindicated eq. 19's "one-liner" approach to Noether's theorem: because the vector field X_f is locally Hamiltonian, it preserves the symplectic structure, i.e. Lie-derives the symplectic form ${\mathcal{L}}_{X_f}\omega =0$ – as a symmetry should.

We also saw in result (2) at the end of Section 3.2.2 that the "meshing", up to a sign, of the Poisson bracket on scalars with the Lie bracket on vector fields implied that the locally Hamiltonian vector fields form a Lie subalgebra of the Lie algebra χ(M) of all vector fields.

Turning to the context of Poisson manifolds, we need to note two points. The first is a similarity with the symplectic case; the second is a contrast.

(1)

One readily checks, just by applying eq. 313 , that the infinitesimal Poisson automorphisms are closed under the Lie bracket. So we write the Lie algebra of these vector fields as P (M): P (M) ⊂ X (M).

(2)

On the other hand, Section 2.1.3's equivalence between a vector field being locally Hamiltonian and preserving the geometric structure of the state-space breaks down.

Agreed, the first implies the second: a locally Hamiltonian vector field preserves the Poisson bracket. We noted this already in Section 5.3.2. The differential statement was that such a field X_H Lie-derives the Poisson tensor: L_XH B^# =0 (eq. 285). The finite statement was that the flows of such a field are Poisson maps: ${\phi }_{\tau }^{*}\{F,G\}=\{{\phi }_{\tau }^{*}F,{\phi }_{\tau }^{*}G\}$ (eq. 284).

But the converse implication fails: an infinitesimal Poisson automorphism on a Poisson manifold need not be locally Hamiltonian. For example, make ℝ² a Poisson manifold by defining the Poisson structure

(314) $\left\{F,H\right\}=x\left(\frac{\partial F}{\partial x}\frac{\partial H}{\partial y}-\frac{\partial H}{\partial x}\frac{\partial F}{\partial y}\right);$

then the vector field $X=\partial /\partial y$ in a neighbourhood of a point on the y-axis is a non-Hamiltonian infinitesimal Poisson automorphism.

This point will affect the formulation of Noether's theorem for Poisson manifolds, in Section 6.2.

Nevertheless, we shall from now on be interested in cases where for all ξ, ξ _M is globally Hamiltonian. This means there is a map J: g → F (M) such that

(315) $X_{J\left(\xi \right)}={\xi }_M$

for all ξ ∈ g. There are three points we need to note about this condition.

(1)

Since the right hand side of eq. 315 is linear in ξ, we can require such a J to be a linear map. For given any J obeying eq. 315, we can take a basis e ₁, …, e_m of g and define a new linear $\overline{j}$ by setting, for any ξ = ξ ⁱ e_i, $\overline{j}$ (ξ):= ξ ⁱ J(e_i ).

(2)

Eq. 315 does not determine J (ξ). For by the linearity of the map B: dJ (ξ) → X_{J (ξ)}, we can add to such a J (ξ) any distinguished function, i.e. an F: M → ℝ such that X_F = 0. That is: X_{J (ξ)+F} ≡ X_{J (ξ)}. (Of course, in the symplectic case, the only distinguished functions are constants.)

(3)

It is worth expressing eq. 315 in terms of Poisson brackets. Recalling that for any F, H ∈ F (M), we have X_H (F) = {F, H}, this equation becomes

(316) $\begin{array}{cc}\left\{F,J\left(\xi \right)\right\}={\xi }_M\left(F\right),\forall F\in \mathcal{F}\left(M\right),& \forall \xi \in \mathcal{g}\end{array}\cdot$

We will also need the following result:

(317) $X_{J\left(\left[\xi ,\eta \right]\right)}=X_{\left\{J\left(\xi \right),J\left(\eta \right)\right\}M}\cdot$

To prove this, we just apply two previous results, each giving a Lie algebra anti-homomorphism.

(i)

Result (4) at the end of Section 4.4: for any left action of Lie group G on any manifold M, the map ξ ↦ ξ _M is a Lie algebra anti-homomorphism between g and the Lie algebra ξ _M of all vector fields on M:

(318) $\begin{array}{cc}{\left(a\xi +b\eta \right)}_M=a{\xi }_M+b{\eta }_M;\left[{\xi }_M,{\eta }_M\right]=-{\left[\xi ,\eta \right]}_M& \forall \xi ,\eta \in \mathcal{g},\text{\hspace{0.17em}and}a,b\in \text{ℝ}\end{array}$

(ii)

The "meshing" up to a sign, just as in the symplectic case, of the Poisson bracket on scalars with the Lie bracket on vector fields, as in eq. 242 at the end of Section 5.2.2:

(319) $X_{\left[F,H\right]}=-\left[X_F,X_H\right]=\left[X_H,X_F\right]\cdot$

So for a Poisson manifold M, the map F ∈ F (M) ↦ X_F ∈ χ (M) is a Lie algebra anti-homomorphism.

Applying (i) and (ii), we deduce eq. 317 by:

(320) $X_{J\left(\left[\xi ,\eta \right]\right)}={\left[\xi ,\eta \right]}_M=-\left[{\xi }_M,{\eta }_M\right]=-\left[X_{J\left(\xi \right)},X_{J\left(\eta \right)}\right]=X_{\left\{J\left(\xi \right),J\left(\eta \right)\right\}M}\cdot$

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QUANTUM STATISTICAL PHYSICS

Gérard G. Emch , in Philosophy of Physics, 2007

REMARKS 56.

1.

Let $\mathcal{A}$ be a C *–algebra, β > 0 and τ be a group of automorphisms of $\mathcal{A}$ . The set $\mathcal{S}$ _β of all KMS states on $\mathcal{A}$ that satisfy the KMS condition for τ and β is convex, i.e. for any two KMS states ψ and χ on $\mathcal{A}$ , with respect to the same τ and β, and any λ ∈ (0, 1): φ = λψ + (1 – Λ)χ is again a KMS state for τ and β.

2.

The set $\mathcal{S}$ _β is closed in the w*– topology it inherits from $\mathcal{A}$ , and it is bounded in the metric topology. Hence it is w*–compact, and the Krein–Milman theorem entails that $\mathcal{S}$ _β is the w*–closed convex hull of the set $\mathcal{S}$ _β of its extreme points [Dunford and Schwartz, 1964, theorem V.8.4]. This ensures not only the existence of extremal points, but also that there are sufficiently many of them: every element in $\mathcal{S}$ _β is the limit of finite convex sums of elements in $\mathcal{S}$ _β; see definition 57 below.

3.

Moreover β₁ ≠ β₂ entails $\mathcal{S}$ _β1 ∩ $\mathcal{S}$ _β2=∅. Incidentally, the GNS representations constructed from states φ₁ ∈ $\mathcal{S}$ _β1 and φ₂ ∈ $\mathcal{S}$ _β2 with β₁ ≠ β₂ are disjoint in the sense that no subrepresentation of one of these is unitarily equivalent to any subrepresentation of the other; cf. [Takesaki, 1970c].

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Loop Groups

Andrew Pressley , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

I.B Twisted Loop Groups

Geometrically, it LG can be thought of as the space of smooth sections of the trivial principal G-bundle on S ¹. More generally, if α is any automorphism of G, one can form a G-bundle on S ¹ by taking the quotient of G  × $\mathbb{R}$ by the equivalence relation that identifies (x, t) with (α(x), t  +   2π) for all x  ∈ G. The cross sections of this bundle form a group L _α G called a twisted loop group. It is clear that L _α G depends only on the class of α modulo inner automorphisms of G, and hence may be assumed to be of finite order if G is semi-simple. Since all G-bundles on S ¹ are trivial if G is connected, any twisted loop group is isomorphic to an untwisted loop group as an abstract group. However, there are good reasons for treating them separately.

Twisted loop groups arise naturally in the study of maximal Abelian subgroups of LG. If T is a maximal torus in G, then LT is obviously a maximal Abelian subgroup of LG. More generally, if τ is any smooth loop in the space of maximal tori of G there is a maximal Abelian subgroup consisting of the loops f such that f(θ)   ∈   τ(θ) for all θ. Up to isomorphism, it depends only on the free homotopy class of τ. The space of maximal tori is G/N(T), where N(T) is the normalizer of T in G. Its fundamental group is the Weyl group W  = N(T)   / T of G. Hence, there is a maximal Abelian subgroup associated to every conjugacy class in W. The group associated to w  ∈ W is the twisted loop group of T associated to the automorphism of T induced by conjugation with a representative of w in N(T).

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Algebra, Abstract

KiHang Kim , Fred W. Roush , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.F Symmetry

An algebraic structure on a set S is a family of subsets $\mathfrak{F}$ _α indexed on α   ∈ I of S ₁  = S ∪ (S  × S) ∪ (S  × S  × S) ∪ ⋯. An automorphism of this structure is a 1–1 onto mapping f  : S  → S such that for all α   ∈ I, T  ∈ $\mathfrak{F}$ _α if and only if f(T)   ∈ $\mathfrak{F}$ _α. For operational structures, this is any 1–1 mapping that is a homomorphism: f(xy)   = f(x)f(y). The complex number system has the automorphism a  + bi  → a  − bi. The additive real numbers has the automorphism x  →   −x.

Especially in a geometric setting, automorphisms are called symmetries. A metric symmetry of a geometric figure is an isometry (distance-preserving map) on its points, which also gives a 1–1 correspondence on its lines and other distinguished subsets. Finite groups of isometries in three-dimensional space include the isometry groups of a regular n-gon (dihedral), cube and octahedron (extension of Z ₂ by $\mathcal{T}$ ₄), icosahedron and dodecahedron (extension of Z ₂ by $\mathcal{A}$ ₅).

The possible isomorphism types of n-dimensional symmetric groups, finite or continuous, are limited. One of the principal applications of group theory is to make use of exact or approximate symmetries of a system in studying it.

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THE REPRESENTATION OF TIME AND CHANGE IN MECHANICS

Gordon Belot , in Philosophy of Physics, 2007

REMARK 39 (Lagrangians and the Reduced Space of Solutions).

In the very simple particle theory considered in example 37 we saw a case in which the reduced space of solutions of a theory admitting gauge symmetries inherited from the original theory a Lagrangian that encoded the gauge-invariant aspects of the original dynamics. But in the more interesting case of Maxwell's theory, considered in example 38, it seems less likely that there is any sense in which the reduced space of solutions arises directly from a local Lagrangian, without passing through a formulation admitting gauge symmetries. And this seems very unlikely indeed if we choose our spacetime to be topologically nontrivial, because in this case the Maxwell field appears to involve a non-local degrees of freedom.

Note that things become even worse in non-Abelian Yang-Mills theories. In these theories, the space of fields is the space of connection one-forms on a suitable principal bundle P → V over spacetime, the Lagrangian is a direct generalization of the Lagrangian of Maxwell's theory, and the group of gauge symmetries is the group of vertical automorphisms of P. The reduced space of solutions is the space of connections modulo vertical automorphisms of P. Even when V is Minkowski spacetime, the best parameterization of the reduced space of solutions would appear to be one that deals with holonomies around closed curves in spacetime. ¹²³ So it would again appear difficult (perhaps impossible) to capture this reduced space of solutions via the variational problem of a local Lagrangian. ¹²⁴ Indeed, it seems plausible the prevalence of gauge freedom in physical theories is grounded in the fact that by including nonphysical variables one is sometimes able to cast an intrinsically nonlocal theory in to a local form. ¹²⁵

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SYMMETRIES AND INVARIANCES IN CLASSICAL PHYSICS

Katherine Brading , Elena Castellani , in Philosophy of Physics, 2007

8.5 Symmetries, objectivity, and objects

Turning now to the issue of the structures left invariant by symmetry transformations, the old and natural idea that what is objective should not depend upon the particular perspective under which it is taken into consideration is reformulated in the following group theoretical terms: what is objective is what is invariant with respect to the relevant transformation group. This connection between symmetries and objectivity is something that has a long history going back to the early twentieth century at least. It was highlighted by Weyl [1952] , where he writes that 'We found that objectivity means invariance with respect to the group of automorphisms.' This connection between objectivity and invariance was discussed particularly in the context of Relativity Theory, both Special and General. We recall Minkowski's famous phrase ([1908] 1923, 75) that 'Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality', following his geometrization of Einstein's Special Theory of Relativity, and the recognition of the spacetime interval (rather than intervals of space and of time) as the geometrically invariant quantity. The connection between objectivity and invariance in General Relativity was discussed by, amongst others, Hilbert and Weyl, and continues to be an issue today. ⁸⁵

Related to this is the use of symmetries to characterize the objects of physics as sets of invariants. Originally developed in the context of quantum theory, this approach can also be applied in classical physics. ⁸⁶ The basic idea is that the invariant quantities — such as mass and charge — are those by which we characterize objects. Thus, through the application of group theory we can use symmetry considerations to determine the invariant quantities and "construct" or "constitute" objects as sets of these invariants. ⁸⁷

In conclusion, then, the philosophical questions associated with symmetries in classical physics are wide-ranging. What we have offered here is nothing more than an overview, influenced by our own interests and puzzles, which we hope will be of service in further explorations of this philosophically and physically rich field.

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Four-dimensional hologram interferometry for automatic detection of oil spills

Maged Marghany , in Synthetic Aperture Radar Imaging Mechanism for Oil Spills, 2020

14.7 Hamming graph for 4D formation from quantum hologram interferometry

Let us deduce that the vertices of the hypercube are V  ∈ H _C , where H _C is the hypercube. In this regard, the hypercube graph is G _{H C} , which is finite. The simplification expression of the graph is G _{H C}   =   (V, E _V ), which is rooted in the adjacency matrix M _A as:

(14.27) $M_{A}G\left[u,v\right]=\left[\left[\left(u,v\right)\in E_V\right]\right]$

In fact, V is a set of vertices and a set of edges E _V between these vertices, where each edge is an unordered pair of vertices. In this circumstance, G _{H C} is required to be vertex-transitive, that is, for any a, b ∈ V, there is an automorphism π  ∈ M _A ut(G _{H C} ) with π(a)   = b π(a)   =   b. The Cartesian product G _{H C}   ⊕ H _m of graphs hypercube graph G _{H C} and Hamming graph H _m (Fig. 14.12), which presents a graph whose adjacency matrix is:

Fig. 14.12. Hamming graph with H _m (3,3).

(14.28) $M\otimes M_A{G}_{H_C}+M_A{H}_m\otimes M$

Let G _{V n} designate the wide-ranging graph on n vertices. Then, the binary n-dimensional hypercube H _{C n} may be defined recursively as:

(14.29) $H_{C_n}=H_{C_n}-1\oplus G_{V_n}\text{For}n\ge 2,\text{and}{H}_{C1}=G_{V2}$

A quantum walk on G _{H C}   ⊕ H _m starting at the vertex (g, h) satisfies:

(14.30) $G_{H_C}\oplus H_m\left|{\varphi }_{G_{H_C}\oplus H_m}\left(t\right)\right.〉=\left|{\varphi }_{G_{H_C}}\left(t\right)\right.〉\otimes \left|{\varphi }_{H_m}\left(t\right)\right.〉$

where ϕ _{G H C} and ϕ _{H m} are quantum phase unwrapping walks on G _{H C} and H _m starting on vertices g and h, respectively. The adjacency matrix G _{H C}   ⊕ H _m is given by:

(14.31) $M_A\otimes H_m+G_{H_C}\otimes M_A$

Because M _A   ⊗ H _m and G _{H C}   ⊗ M _A transform, then the quantum phase unwrapping walks are formulated as:

(14.32) $\left|{\varphi }_{G_{H_C}\oplus H_m}\left(t\right)\right.〉=\left|{\varphi }_{G_{H_C}}\left(t\right)\right.〉\otimes \left|{\varphi }_{H_m}\left(t\right)\right.〉$

In quantum mechanics, the natural approach to indicate dual systems is across the tensor product ⊗. In this understanding, the Cartesian graph artifact intends a comparable function for quantum walks of the phase unwrapping. Eq. (14.32) is the key ingredient to constructing hypercube hologram interferometry through the Hamming graphs. In general, the Hamming graphs deliver tight characterization of quantum uniform mixing [12].

Therefore, the propagator can consistently be assumed as a recitation of a quantum random walk on a systematic graph involving n-dimensional hypercube having a self-loop edge devoted separately to its vertices. The final version of the quantum walk for optimal search 2r can, therefore, be expressed by an alternating sequence of the unitary operators U′′ _m ⁽⁺⁾ as:

(14.33) $\left|{{\varphi }^{\left(e\right)}}_{G_{H_C}\oplus H_m}\left(2r\right)\right.〉={\left(U_m^{\left(+\right)},U_m^{\prime \prime \left(+\right)}\right)}^r\left|{{\varphi }^e}_{G_{H_C}}\left(t\right)\right.〉\otimes \left|{\varphi }_{H_m}\left(t\right)\right.〉$

where

(14.33.1) $U_m^{\prime \prime \left(+\right)}=S^{\left(+\right)}{C}^{\prime \prime \left(+\right)}$

Here C′′⁽⁺⁾ is the coin operator, which acts on the total Hilbert space Η ^C   ⊗   Η ^V and the S is a propagator of a quantum walk across the hypercube vertices V as:

(14.33.2) $S^{\left(+\right)}=\sum _{d,\overrightarrow{x}}\left|\left|d,\overrightarrow{x}\oplus {\overrightarrow{e}}_d\right.〉\left|d,\overrightarrow{x}\right.〉\right|$

Here d is the direction of propagation, $\overrightarrow{e}$ is the edge of the vertices, and ⊕ denotes the bit-wise addition modulo 2 operator. Moreover, $\overrightarrow{x}$ is the Hamming weight of an integer, which is the number of 1   s in its binary string.

The coin operator C′′⁽⁺⁾ is then determined from:

${{C^{\prime }}^{\prime }}^{\left(+\right)}=C_0\otimes M_A+\left(C_1-C_0\right)\otimes \sum _{j=1}^m\left|{\overrightarrow{x}}_{\mathit{tg}}^{\left(j\right)}\right.〉\left.〈{\overrightarrow{x}}_{\mathit{tg}}^{\left(j\right)}\right|$

C ₀ presents the n-dimensional Grover operator, which is also known as the "Grover diffusion operator," and C ₁ is chosen to be −   1. However, due to the symmetry of the hypercube graph (Fig. 14.13), the vertices can always be relabeled in such a way that the marked vertex becomes ${\overrightarrow{x}}_{\mathit{tg}}^{\left(j\right)}=0$ . To formalize the task of finding multiple marked vertices, let us denote the number of elements marked by the oracle by m, and their labels by ${\overrightarrow{x}}_{\mathit{tg}}^{\left(j\right)}$ and j  =   1, …, m [12].

Fig. 14.13. 3D encoded into a 4D hypercube.

Fig. 14.13 explains how 3D is encoded into a 4D hypercube. The effect of operator X is switching between phase unwrapping and vertices. In other words, the processors in the cubes of dimension 1, 2, and 3 are categorized with integers, which are represented as binary numbers. Therefore, those dual processors are neighbors in dimension ${\overrightarrow{x}}_{\mathit{tg}}^{\left(j\right)}$ in a hypercube of dimension, and a message can be routed between any pair of processors at most X hops.

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Group Theory

Ronald Solomon , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III Basic Constructions

In Section I we addressed the problem of decomposing a group into its constituent simple composition factors (when possible). Now we consider the opposite problem of "composing" two or more groups to create a new and larger group. One fundamental construction is the direct product construction.

Definition:

Let G and H be groups. The Cartesian product set G   ×   H with the operation

$(\text{g},\text{h})\cdot ({\text{g}}^{\prime },{\text{h}}^{\prime })=(\text{g}{\text{g}}^{\prime },\text{h}{\text{h}}^{\prime })$

is a group, called the direct product G   ×   H.

This construction can be iterated to define direct products ∏ G _i over arbitary index sets I. Another important generalization is the semidirect product. Before defining this, we must generalize the notion of an isomorphism of groups.

Definition:

Let (G, ∘) and (H, ·) be groups. A function f: G   →   H is a homomorphism of group provided that

$\begin{array}{ll}\text{f}(\text{g}\circ {\text{g}}^{\prime })=\text{f}(\text{g})\cdot \text{f}({\text{g}}^{\prime })\hfill & \mathrm{for\; all}\hfill \end{array}\text{g},{\text{g}}^{\prime }\in \text{G}.$

Thus an isomorphism of groups is a bijective homomorphism of groups. In general, failure of injectivity for homomorphisms is measured by the following subgroup.

Definition:

Let f: G   →   H be a homomorphism of groups. The kernel of f is defined by

$\text{Ker}\left(\text{f}\right)=\left\{\text{g}\in \text{G}:\text{f}\left(\text{g}\right)={\text{e}}_{\text{H}}\right\},$

where e _H   is the identity element of the group H.

The pre-image f ⁻¹(h) of each element in f(G) is both a left and right coset of Ker(f)   = f ⁻¹(e _H ) and so has the same cardinality as Ker(f). In particular, f is injective if and only if Ker(f)   =   {e _G }, where e _G is the identity element of G. There is an intimate connection between normal subgroups and homomorphisms.

Theorem:

Let f: G   →   H be a homomorphism of groups with kernel K. Then K is a normal subgroup of G and f induces an isomorphism between G/K and f(G). Conversely, if K is any normal subgroup of the group G, then the function π: G   →   G/K defined by

$\begin{array}{ll}\pi (\text{g})=g\text{K}\hfill & \mathrm{for\; all}\hfill \end{array}\text{g}\in \text{G}$

is a homomorphism of groups with kernel K.

Now we can describe the semidirect product construction of groups.

Definition:

Let K and H be groups and let ϕ: H  →   Aut(K) be a homomorphism of groups. The semidirect product K: ^ϕ H (or simply K: H) is the group whose underlying set is the Cartesian product set K   ×   H with multiplication defined by

$\begin{array}{lll}(\text{k},\text{h})({\text{k}}^{\prime },{\text{h}}^{\prime })\hfill & =\hfill & (k\varphi (\text{h})({\text{k}}^{\prime }),\text{h}{\text{h}}^{\prime })\hfill \\ \hfill & \hfill & \mathit{for\; all}\text{k},{\text{k}}^{\prime }\in \text{K},\text{h},{\text{h}}^{\text{'}}\in \text{H}\hfill \end{array}$

The direct product K  × H is the special case of the above construction in which ϕ is the homomorphism mapping every element of H to the identity automorphism of K. In all cases, {(k, e _H ):k  ∈ K} is a normal subgroup of K: H, which is isomorphic to K. The subset H ₀  =   {(e _K , h):h  ∈ H} is a subgroup of K: H isomorphic to H, but is not in general a normal subgroup of K: H. Indeed, H ₀ is also normal if and only if K: H  ≅   K  × H.

If all composite (i.e., not simple) groups could be constructed from proper subgroups by an iterated semidirect product construction, then the classification of all finite groups, or even all groups having a composition series, would at least be thinkable, if not doable. However, not, all composite groups can be so constructed, as is illustrated by the easy example of the cyclic group C _{p ²} . The obstruction to a composite group "splitting" as a semidirect product was first analyzed by Schur. The attempt to parametrize the set of all groups which are an "extension" of a given normal subgroup by a given quotient group was one of the motivating forces for the development of the cohomology theory of groups.

In the context of finitely generated abelian groups, a complete classification is nevertheless possible and essentially goes back to Gauss. (A group G is finitely generated if there is a finite subset S of G such that every element of G is expressible as a finite word in the "alphabet" S. We write G  =   〈S〉 if G is generated by the elements of the subset S. In particular, every finite group G is finitely generated: you may take S  = G.)

Theorem:

If A is a finitely generated abeliean group, then A is a finite direct product of cyclic groups.

On the other hand, the enumeration of all finite p-groups is an unthinkable problem. For p a prime, we call a group P a p-group if all of the elements of P have order a power of p. By theorems of Lagrange and Cauchy, P is a finite p-group if and only if ∣P∣   = p ⁿ for some n  ≥   0. G. Higman and Sims have established that the number P(n) of p-groups of cardinality p ⁿ is asymptotic to a function of the form ${\text{p}}^{\frac{2}{27}{\text{n}}^3+\text{o}({\text{n}}^3)}$ as n  →   ∞.

In a different direction, another important generalization of the direct product is the central product. First we introduce an important normal subgroup of a group G.

Definition:

Let G be a group. The center, Z(G), of G is defined by

$\text{Z}\left(\text{G}\right)=\left\{\text{z}\in \text{G}:z\text{g}=\text{gz}\text{f}\text{o}\text{r}\text{a}\text{l}\text{l}\text{g}\in \text{G}\right\}.$

Definition:

Let H, K, and C be groups and let α: C   →   Z(H) and β: C   →   Z(K) be injective homomorphisms. Let

$\text{Z}=\left\{\left(\alpha \left(\text{c}\right),\beta \left(\text{c}\right)\right)\in \text{H}\times \text{K}:\text{c}\in \text{C}\right\}.$

Then the central product H  ∘ _C K is the quotient group (H   ×   K)/Z.

Sometimes the central product is (ambiguously) written H   ∘   K. Often this is done when Z(H)   ≅ Z(K) and α and β are understood to be isomorphisms. Like the direct product, all of these product constructions can be iterated.

In some ways dual to the center of G is the commutator quotient.

Definition:

Let G be a group. If x, y   ∈   G, then the commutator [x, y]   =   x⁻¹y⁻¹ xy. The commutator subgroup [G, G] of G is defined by

$[\text{G},\text{G}]=〈[\text{x},\text{y}]:\text{x},\text{y}\in \text{G}〉$

i.e., [G, G] is the subgroup of G generated by all commutators of elements of G. Then [G, G] is a normal subgroup of G and the commutator quotient G  /   [G, G] is the largest abelian quotient group of G.

Note that G is an abelian group if and only if Z(G)   = G if and only if [G, G]   =   {e}. Sometimes G  /   [G, G] is called the abelianization of G.

This leads to the definition of the following important classes of groups.

Definition:

A nonidentity finite group G is quasi-simple if G  =   [G, G] and G/Z(G) is a (nonabelian) simple group. A finite group G is semisimple if G   =   G ₁  ∘ G ₂  ∘   …   ∘ G _r , where each G _i is a quasisimple group.

In the contexts of Lie groups and algebraic groups, a group is called simple if every proper closed normal subgroup is finite. Thus the group SL(n, C), whose center is isomorphic to the finite group of complex nth roots of 1, would be called a simple group by a Lie theorist and a quasi-simple (but not simple) group by a finite group theorist. There is a notion of semisimple group in the categories of linear algebraic groups and Lie groups which coincides with the definition above for connected groups.

Some slightly larger classes of groups play an important role in many areas. The following definitions are not standard. The concept of a connected reductive group is fundamental in the theory of algebraic groups, in which context it is defined to be the product of a normal semisimple group and a torus (a group isomorphic to a product of GL ₁'s). The second definition below approximates this notion in the category of all groups.

Definition:

A group G is almost simple if G has a normal subgroup E which is quasi-simple and G/Z(E) is isomorphic to a subgroup of (Aut)(E). A group G is almost semisimple if G has a normal subgroup E  = S  ∘ Z(E) with S   =   S ₁  ∘ S ₂  ∘   …   ∘ S _r semisimple (with each  S _i quasi-simple)   and G/Z(E) is isomorphic to a subgroup of (Aut)(S) normalizing each S _i .

Most of the classical groups described above are almost semisimple. Indeed, S is a quasi-simple group in most cases. Also, the Levi complements of parabolic subgroups (stabilizers of flags of totally singular subspaces) of the classical linear groups are usually almost semisimple, exceptions arising over small fields or because of the peculiar structure of certain four-dimensional orthogonal groups. In a vast number of cases of importance in mathematics and the physical sciences, the symmetry group (or automorphism group) of an interesting structure is either an almost semisimple group or has the form G  =   V: H as the semidirect product of an abelian group V and an almost semisimple group H. For example, the group of all rigid motions of Euclidean space R ⁿ has the structure R ⁿ : O(n). Similarly, if V is an affine space, then the group AGL(V) of all affine transformations of V has the structure AGL(V)   = V:GL(V).

The proof of the classification of finite simple groups necessitated in fact the classification of all finite almost-simple groups. Thus, as a consequence of the classification (along with the classification of finite abelian groups), there is essentially a complete description of all finite almost-semisimple groups. The analogous result for infinite groups is out of reach, but if one restricts to the categories of algebraic groups or Lie groups, then a description of all connected reductive groups is again a part of the fundamental classification theorems.

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