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I think this is the Baker-Campbell-Haussdorff formula, and the notation means to iterate the commutator. The notation is short-hand for an expression utilizing the Backer Campbell Haussdorf formula. This is not standard notation, and one would typically expect any text that uses it to define it at its first occurrence.

Since you understandably cannot provide us with a reference, your best bet is hunting for all occurrences of that notation, starting from there and going up through the text, until it explains what it means. Trust me, it will be there. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Commutator summation notation Ask Question. Asked 6 years, 6 months ago.

Active 6 years, 6 months ago. Viewed times. See edit for full equation. Active Oldest Votes. Robin Ekman Robin Ekman 13k 1 1 gold badge 25 25 silver badges 50 50 bronze badges.

A reference to this is this section in Wikipedia. JamalS JamalS 17k 5 5 gold badges 40 40 silver badges 94 94 bronze badges.In mathematicsparticularly in linear algebramatrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix, known as the matrix producthas the number of rows of the first and the number of columns of the second matrix.

Matrix multiplication was first described by the French mathematician Jacques Philippe Marie Binet in[3] to represent the composition of linear maps that are represented by matrices. Matrix multiplication is thus a basic tool of linear algebraand as such has numerous applications in many areas of mathematics, as well as in applied mathematicsstatisticsphysicseconomicsand engineering.

This article will use the following notational conventions: matrices are represented by capital letters in bold, e. Avectors in lowercase bold, e. A and a. Index notation is often the clearest way to express definitions, and is used as standard in the literature. The i, j entry of matrix A is indicated by A ijA ij or a ijwhereas a numerical label not matrix entries on a collection of matrices is subscripted only, e.

Thus the product AB is defined if and only if the number of columns in A equals the number of rows in B[2] in this case n. In most scenarios, the entries are numbers, but they may be any kind of mathematical objects for which an addition and a multiplication are defined, that are associativeand such that the addition is commutativeand the multiplication is distributive with respect to the addition.

**Quantum Operators**

In particular, the entries may be matrices themselves see block matrix. The figure to the right illustrates diagrammatically the product of two matrices A and Bshowing how each intersection in the product matrix corresponds to a row of A and a column of B. Historically, matrix multiplication has been introduced for facilitating and clarifying computations in linear algebra.

This strong relationship between matrix multiplication and linear algebra remains fundamental in all mathematics, as well as in physicsengineering and computer science.

## Quantum Mechanics/Operators and Commutators

If a vector space has a finite basisits vectors are each uniquely represented by a finite sequence of scalars, called a coordinate vectorwhose elements are the coordinates of the vector on the basis. These coordinate vectors form another vector space, which is isomorphic to the original vector space.

A coordinate vector is commonly organized as a column matrix also called column vectorwhich is a matrix with only one column. So, a column vector represents both a coordinate vector, and a vector of the original vector space. A linear map A from a vector space of dimension n into a vector space of dimension m maps a column vector. The linear map A is thus defined by the matrix.

Korg new synthThe general form of a system of linear equations is. Using same notation as above, such a system is equivalent with the single matrix equation. The dot product of two column vectors is the matrix product. More generally, any bilinear form over a vector space of finite dimension may be expressed as a matrix product.

Matrix multiplication shares some properties with usual multiplication. However, matrix multiplication is not defined if the number of columns of the first factor differs from the number of rows of the second factor, and it is non-commutative[10] even when the product remains definite after changing the order of the factors. Therefore, if one of the products is defined, the other is not defined in general.

Even in this case, one has in general. If, instead of a field, the entries are supposed to belong to a ringthen one must add the condition that c belongs to the center of the ring.

The matrix product is distributive with respect to matrix addition. If the scalars have the commutative property, then all four matrices are equal. These properties result from the bilinearity of the product of scalars:.

If the scalars have the commutative propertythe transpose of a product of matrices is the product, in the reverse order, of the transposes of the factors.

That is. This identity does not hold for noncommutative entries, since the order between the entries of A and B is reversed, when one expands the definition of the matrix product. If A and B have complex entries, then.Commutators and conjugates are very simple ways of building sequences of moves that can, for example, permute pieces or orientate them. Before getting started, you should know, at least, how to solve the cube by any method and maybe, have basic intuition in solving other cube variants.

A three-cycle is a rotation of the place of three pieces. They have to be obviously of the same kind. For instance, three corners or three edges. In this kind of commutator, X will be any moves you need to do to insert the piece which does not belong to the reference level in its objective place within it.

This place has to be one of the two pieces you are involving in the three-cycle, obviously. This algorithm cannot modify anything in the reference level except for the change of one piece for another. Y will be whatever move or moves you need, in order to put the piece in the reference level you have not yet exchanged, to that slot you just inserted a piece using algorithm X. What we will do with algorithm X is to take one of the pieces out and eventually place it back in its right orientation.

Remember that you cannot modify anything else from the reference level. This kind of commutator swaps two pairs of two pieces. They are not very useful because in many cases you will just need a bunch of three-cycles to solve a big part of most of the cubes.

Even though, they are very easy to work out and may be useful sometimes. A conjugate is the perfect tool you need if you have come up with a commutator. They are very simple; you first do the moves you need to do to place the pieces you want to swap in the desired place or orientation. It can completely mess up your cube. Then, you execute the algorithm and afterwards, you undo the moves you did in the first place.

If you want to do a three-cycle and orientate the pieces, you can do both at the same time using a conjugate.Forums New posts Search forums. Media New media New comments Search media. Members Current visitors New profile posts Search profile posts.

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### Commutators and Conjugates

For a better experience, please enable JavaScript in your browser before proceeding. Help with commutator notation? Thread starter G Start date Jul 23, Tags commutators conjugates notation. G Member. Thanks for your time. No, it can't be done with any algorithm, because most algorithms don't have the repeating patterns like setups or inverses. You should probably specify "that decomposition". If you give me any algorithm A1 A2 A An with Ai being any turn i can "decompose" that into [A1:A1] How would you do it?

I know that for example this 3-cycle R U R' D R U' R' D' can be written as [R U R', D], and I also know that the algorithm above T perm also can be written with that notation, but I don't know how to "translate" from one notation standard to the other commuator notation.

Is there a method? Or you just have to think? Stefan Member. G said:. Stefan said:. Ranzha Friendly, Neighbourhoodly.Instead we use operators. Now this could look deceptively simple if we didn't use operators for energy and momentum.

But like most things, it's never simple. The energy operator acts on the wave function, as does the momentum operator. So we need to find the wave function in order to make any sense of this equation. Although we could theoretically come up with an infinite number of operators, in practice there are a few which are much more important than any others.

In Quantum Mechanics, everything is probabilistic e. So we often want to know the expected value of position, momentum, or anything else, and there is quite a nice method of doing this.

So, for instance, if you knew the wave function and wanted to find the expectation value of momentum you'd use the equation. This notation is very widely used.

Buzzbreak pcIt is extremely useful in finite dimensional problems, represented by matrices and state vectors. In 2D, qubits are used most used in Quantum Information Theorywhere:. Commutators are very important in Quantum Mechanics. As well as being how Heisenberg discovered the Uncertainty Principle, they are often used in particle physics.

It is known that you cannot know the value of two physical values at the same time if they do not commute. Essentially, this implies that you cannot simultaneously know position and momentum precisely for a given moment in time. From this, we know that momentum and energy commute.

Thus, we can find simultaneous eigenfunctions of energy and momentum with definite values of the two observables. From Wikibooks, open books for an open world.

Quantum Mechanics. Category : Book:Quantum Mechanics.

Namespaces Book Discussion. Views Read Edit View history.In mathematicsthe commutator gives an indication of the extent to which a certain binary operation fails to be commutative.

There are different definitions used in group theory and ring theory. The commutator of two elements, g and hof a group Gis the element. The set of all commutators of a group is not in general closed under the group operation, but the subgroup of G generated by all commutators is closed and is called the derived group or the commutator subgroup of G. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as.

Libra depressionCommutator identities are an important tool in group theory. It is a group-theoretic analogue of the Jacobi identity for the ring-theoretic commutator see next section. Similar identities hold for these conventions. Many 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:. If the derived subgroup is central, then.

The commutator of two elements a and b of a ring including any associative algebra is defined by. It is zero if and only if a and b commute. In linear algebraif two endomorphisms of a space are represented by commuting matrices in terms of one basis, then they are so represented in terms of every basis.

By using the commutator as a Lie bracketevery associative algebra can be turned into a Lie algebra. The anticommutator of two elements a and b of a ring or an associative algebra is defined by. The commutator of two operators acting on a Hilbert space is a central concept in quantum mechanicssince it quantifies how well the two observables described by these operators can be measured simultaneously.

Relation 3 is called anticommutativitywhile 4 is the Jacobi identity. In other words, the map ad A defines a derivation on the ring R. Identities 23 represent Leibniz rules for more than two factors, and are valid for any derivation. Identities 4 — 6 can also be interpreted as Leibniz rules. Identities 78 express Z - bilinearity. When dealing with graded algebrasthe commutator is usually replaced by the graded commutatordefined in homogeneous components as.

Especially if one deals with multiple commutators in a ring Ranother notation turns out to be useful. This mapping is a derivation on the ring R :. By the Jacobi identityit is also a derivation over the commutation operation:. The general Leibniz ruleexpanding repeated derivatives of a product, can be written abstractly using the adjoint representation:.

From Wikipedia, the free encyclopedia. This article is about the mathematical concept. For the electrical component, see Commutator electric.In mathematicsmore specifically in abstract algebrathe commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian.

So in some sense it provides a measure of how far the group is from being abelian; the larger the commutator subgroup is, the "less abelian" the group is. Here are some simple but useful commutator identities, true for any elements sgh of a group G :.

The first and second identities imply that the set of commutators in G is closed under inversion and conjugation. However, the product of two or more commutators need not be a commutator.

It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96; in fact there are two nonisomorphic groups of order 96 with this property. This shows that the commutator subgroup can be viewed as a functor on the category of groupssome implications of which are explored below. For a finite group, the derived series terminates in a perfect groupwhich may or may not be trivial.

For an infinite group, the derived series need not terminate at a finite stage, and one can continue it to infinite ordinal numbers via transfinite recursionthereby obtaining the transfinite derived serieswhich eventually terminates at the perfect core of the group. The abelianization functor is the left adjoint of the inclusion functor from the category of abelian groups to the category of groups.

Equivalently, if and only if the group equals its abelianization. See above for the definition of a group's abelianization. Equivalently, if and only if the abelianization of the group is trivial. This is "opposite" to abelian. Since the derived subgroup is characteristicany automorphism of G induces an automorphism of the abelianization.

Since the abelianization is abelian, inner automorphisms act trivially, hence this yields a map. From Wikipedia, the free encyclopedia. Smallest normal subgroup by which the quotient is commutative.

Main article: Commutator. Main article: Perfect group.

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