Master of Science in Chemistry

Commutator Bracket In Group Theory.

The commutator of two elements, g and h, of a group, G, is the element

[g, h] = g⁻¹h⁻¹gh.

It is equal to the group's identity if and only if g and h commute (i.e., if and only if gh = hg). The subgroup of $G$ generated by all commutators is called the derived group or the commutator subgroup of G. Note that one must consider the subgroup generated by the set of commutators because in general the set of commutators is not closed under the group operation. Commutators are used to define nilpotent and solvable groups.

N.B. The above definition of the commutator is used by group theorists. Many other mathematicians define the commutator as

[g, h] = ghg⁻¹h⁻¹.

Identities

Commutator identities are an important tool in group theory, (McKay 2000, p. 4). The expression a^x denotes x⁻¹a x.

$x^y = x[x,y].\,$
$[y,x] = [x,y]^{-1}.\,$
$[x y, z] = [x, z]^y\cdot [y, z]$ and $[x, y z] = [x, z]\cdot [x, y]^z.$
$[x, y^{-1}] = [y, x]^{y^{-1}}$ and $[x^{-1}, y] = [y, x]^{x^{-1}}.$
$[[x, y^{-1}], z]^y\cdot[[y, z^{-1}], x]^z\cdot[[z, x^{-1}], y]^x = 1$ and $[[x,y],z^x]\cdot [[z,x],y^z]\cdot [[y,z],x^y]=1.$

Identity 5 is also known as the Hall-Witt identity. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator (see next section).

N.B. The above definition of the conjugate of a by x is used by group theorists. Many other mathematicians define the conjugate of a by x as xax⁻¹. This is often written $x a$ . Similar identities hold for these conventions.

A wide range of identities are used that are true modulo certain subgroups. These can be particularly useful in the study of solvable groups and nilpotent groups. For instance, in any group second powers behave well

$(xy)^2 = x^2y^2[y,x][[y,x],y].\,$

If the derived subgroup is central, then

$(xy)^n = x^n y^n [y,x]^{\binom{n}{2}}.$

Ring theory

The commutator of two elements a and b of a ring or an associative algebra is defined by

[a, b] = ab − ba.

It is zero if and only if a and b commute. In linear algebra, if two endomorphisms of a space are represented by commuting matrices with respect to one basis, then they are so represented with respect to every basis. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. The commutator of two operators defined on a Hilbert space is an important concept in quantum mechanics since it measures how well the two observables described by the operators can be measured simultaneously. The uncertainty principle is ultimately a theorem about these commutators via the Robertson-Schrödinger relati

Identities

The commutator has the following properties:

Lie-algebra relations:

$[A, A] = 0$
$[A, B] = - [B, A]$
$[A,[B, C]] + [B,[C, A]] + [C,[A, B]] = 0$

The second relation is called anticommutativity, while the third is the Jacobi identity.

Additional relations:

$[A, B C] = [A, B] C + B [A, C]$
$[A B, C] = A [B, C] + [A, C] B$
$[A B C, D] = A B [C, D] + A [B, D] C + [A, D] B C$
$[A B, C D] = A [B, C D] + [A, C D] B = A [B, C] D + A C [B, D] + [A, C] D B + C [A, D] B$
$[[[A, B], C], D] + [[[B, C], D], A] + [[[C, D], A], B] + [[[D, A], B], C] = [[A, C],[B, D]]$
$[A B, C] = A {B, C} - {A, C} B$ , where {A,B}=AB+BA is the anticommutator defined below

If $A$ is a fixed element of a ring $\scriptstyle\mathfrak{R}$ , the first additional relation can also be interpreted as a Leibniz rule for the map $\scriptstyle D_A: R \rightarrow R$ given by $\scriptstyle B \mapsto [A,B]$ . In other words: the map $D A$ defines a derivation on the ring $\scriptstyle\mathfrak{R}$ .

The following identity involving commutators, a special case of the Baker-Campbell-Hausdorff formula, is also useful:

$e^{A}Be^{-A}=B+[A,B]+\frac{1}{2!}[A,[A,B]]+\frac{1}{3!}[A,[A,[A,B]]]+...$

Graded rings and algebras

When dealing with graded algebras, the commutator is usually replaced by the graded commutator, defined in homogeneous components as $\ [\omega,\eta]_{gr} := \omega\eta - (-1)^{\deg \omega \deg \eta} \eta\omega.$

Derivations

Especially if one deals with multiple commutators, another notation turns out to be useful involving the adjoint representation:

$\operatorname{ad} (x)(y) = [x, y] .$

Then $ad(x)$ is a derivation and $ad$ is linear, i.e., $ad(x + y) = ad(x) + ad(y)$ and ${\rm ad} (\lambda x)=\lambda\,\operatorname{ad} (x)$ , and a Lie algebra homomorphism, i.e., $ad([x, y]) = [ad(x),ad(y)]$ , but it is not always an algebra homomorphism, i.e. the identity $\operatorname{ad}(xy) = \operatorname{ad}(x)\operatorname{ad}(y)$ does not hold in general.

Examples:

${\rm ad} (x){\rm ad} (x)(y) = [x,[x,y]\,]$
${\rm ad} (x){\rm ad} (a+b)(y) = [x,[a+b,y]\,].$

Anticommutator

The anticommutator of two elements a and b of a ring or an associative algebra is defined by

{a, b} = ab + ba.

The anticommutator is used less often than the commutator, but can be used for example to define Clifford algebras and Jordan algebras.

Poisson Brackets

For functions that are defined on the phase space we can define the following operation. Let F = F(q, p, t) and G = G(q, p, t). Then a Poisson bracket of these two functions is defined by:

$\begin{displaymath} \left\{ F, G \right\} = \sum_{i=1}^n \left( \frac{\pa... ...al G}{\partial q_i} \frac{\partial F}{\partial p_i} \right) \end{displaymath}$

(4.10)

This operation has the following neat properties:

$\displaystyle \left\{ F, G \right\}$	=	$\displaystyle - \left\{ G, F \right\}$	(4.11)
$\displaystyle \left\{ F, F \right\}$	=	0	(4.12)
$\displaystyle \left\{ F_1 + F_2, G \right\}$	=	$\displaystyle \left\{ F_1, G \right\} + \left\{ F_2, G \right\}$	(4.13)
$\displaystyle \left\{ F_1 F_2, G \right\}$	=	$\displaystyle F_1 \left\{ F_2, G \right\} + F_2 \left\{ F_1, G \right\}$	(4.14)
$\displaystyle \left\{ F, q_i \right\}$	=	$\displaystyle - \frac{\partial F}{\partial p_i}$	(4.15)
$\displaystyle \left\{ F, q_i \right\}$	=	$\displaystyle \frac{\partial F}{\partial q_i}$	(4.16)
$\displaystyle \left\{ q_i, q_j \right\}$	=	0	(4.17)
$\displaystyle \left\{ p_i, p_j \right\}$	=	0	(4.18)
$\displaystyle \left\{ q_i, p_j \right\}$	=	$\displaystyle \delta_{ij}$	(4.19)
0	=	$\displaystyle \left\{ F_1, \left\{ F_2, F_3 \right\} \right\} + \left\{ F_2, \left\{ F_3, F_1 \right\} \right\} + \left\{ F_3, \left\{ F_1, F_2 \right\} \right\}$	(4.20)
$\displaystyle \frac{\partial}{\partial t} \left\{ F, G \right\}$	=	$\displaystyle \left\{ \frac{\partial F}{\partial t}, G \right\} + \left\{ F, \frac{\partial G}{\partial t} \right\}$	(4.21)

Poisson brackets can be used to express time derivatives of phase space functions:

$\displaystyle \frac{\textrm{d} F}{\textrm{d} t}$	=	$\displaystyle \sum_{i=1}^n\left(\frac{\partial F}{\partial q_i} \frac{\textrm{d... ...p_i} \frac{\textrm{d} p_i}{\textrm{d} t}\right) + \frac{\partial F}{\partial t}$
	=	$\displaystyle \sum_{i=1}^n\left( \frac{\partial F}{\partial q_i} \frac{\partial... ...al p_i} \frac{\partial H}{\partial q_i} \right) + \frac{\partial F}{\partial t}$
	=	$\displaystyle \left\{F, H\right\} + \frac{\partial F}{\partial t}$	(4.22)

This equation can then be applied to q_i and p_i itself to re-express the Hamilton equations in the following form:

$\displaystyle \frac{\textrm{d} q_i}{\textrm{d} t}$	=	$\displaystyle \left\{ q_i, H \right\}$	(4.23)
$\displaystyle \frac{\textrm{d} p_i}{\textrm{d} t}$	=	$\displaystyle \left\{ p_i, H \right\}$	(4.24)

In turn, substituting H in place of F yields:

$\begin{displaymath}\frac{\textrm{d} H}{\textrm{d} t} = \left\{H, H\right\} + \frac{\partial H}{\partial t} = \frac{\partial H}{\partial t} \end{displaymath}$

(4.25)

Expressions such as $\left\{q_i, p_j\right\} = \delta_{ij}$ ought to tug at the heart of everyone acquainted with Quantum Mechanics, where one of the expressions of the Heisenberg Uncertainty Principle is

$\begin{displaymath}\left[\hat{q}_i, \hat{p}_j\right] = i\hbar\delta_{ij}, \end{displaymath}$

where

$\begin{displaymath}\left[\hat{q}_i, \hat{p}_j\right] = \hat{q}_i \hat{p}_j - \hat{p}_j \hat{q}_i \end{displaymath}$

is a commutator of operators that represent position and momentum. Similarly time evolution of any Quantum Mechanical operator that does not depend on time explicitly is given by

$\begin{displaymath}\left[\hat{\Psi}, \hat{H}\right] = i\hbar \frac{\textrm{d} \hat{\Psi}}{\textrm{d} t} \end{displaymath}$

This is not entirely an accident. Poisson brackets lead directly to the so called canonical quantization. Canonical quantization is a procedure which converts a classical field theory or a classical mechanical theory into the corresponding Quantum theory. One of its rules is:

$\begin{displaymath}\left\{\Psi, \Phi\right\} \rightarrow \frac{1}{i\hbar} \left[\hat{\Psi}, \hat{\Phi}\right] \end{displaymath}$

But the truth about canonical quantization carried out like that is that it has to be interfered with frequently in order to deliver a meaningful Quantum theory, and the reason for that is that Quantum theories cannot be derived formally from classical theories. The opposite is the case, i.e., Quantum theories are a lot richer than classical theories, and it is the latter that are derivable from the former in thermodynamic limit. But canonical quantization was useful in its day in providing a bridge between XIXth century classical physics and XXth century quantum physics

Master of Science in Chemistry

Pages

Wednesday, March 30, 2011

Commutator Bracket Msc 2nd Year

Identities

Ring theory

Graded rings and algebras

Anticommutator

Poisson bracket

Poisson Brackets