User:Terry Bollinger/x1

Original article: Operator (physics)

In physics, an operator is the mathematical abstraction of a real-world activity that either transforms (changes the state) or observes (extracts information about) an object or system. For example, adding heat to an object is a transformation activity, while measuring any ensuing changes in length is an observation activity; both can be represented by operators.

Operators in classical and quantum physics
In classical mechanics, transformation and observation activities were assumed in general to be independent of each other. Consequently, the operators used to represent them mathematically were designed to be similarly independent. (This assumption of independence is not strictly true even within classical mechanics; see chaos theory and turbulence, for example.) Given the assumption that transformation and observation were independent, the general strategy in classical mechanics was to minimize the effects of observation to the point where a measurement activity could be modeled as a null operator — that is, as an operation whose effect is so minimal that it can be eliminated from consideration in the overall mathematical analysis of the system. This proved to be a powerful and effective strategy, since it greatly simplified the overall mathematical structure needed to make accurate predictions about may types of physical behaviors.

With the arrival of quantum mechanics in the early 1900s it was realized it is not in general possible to observe very small objects and systems without simultaneously transforming their states, often dramatically. To create mathematical models capable of dealing with this unexpected complication, physicists were forced to develop new operators capable of modeling both transformation and observation. Many of the more paradoxical aspects of quantum mechanics stem directly from this need to place the activities of transformation and observation on an equal mathematical footing. Another consequence is that even though classical operators provide important clues to the construction of their quantum mechanical equivalents, the quantum mechanical versions necessarily differ in ways that reflect the impact of including the effects of observation as part of their overall structure.

Relationship to operators in mathematics
As implied by its name, the mathematical abstraction used to represent both transformational and observational activities in physics is the same operator (or, equivalently, function) concept encountered in such mundane activities such as addition, subtraction, and trigonometry. The physics concept of an operator "feels" more complicated not because of any additions to the mathematical idea, but because the physical world imposes many highly specific constraints on the mathematical operators used to model it. This is particularly true for quantum mechanics, where operators are used to capture succinctly counterintuitive rules resulting from decades of physics experimentation.

The power of the simple operator notation is that when used properly, it can capture many literally universal rules of physics in the form of brief, powerful, and mathematically convenient notations.

Use of operators to represent symmetry in classical physics
The simplest example of the utility of operators is the study of symmetry. Because of this, they are a very useful tool in classical mechanics. In quantum mechanics, on the other hand, they are an intrinsic part of the formulation of the theory.

Operators provide a mathematically precise way to capture and extend the everyday concept of symmetry. Informally, if one says that an object is "symmetrical" it means that if a certain precisely specified action is performed on it, any observer who sees the object before and after the action will be unable to tell whether anything has changed. An example of such an action is rotating a perfect cube by one quarter of a turn. While an observer brought in before and after the action may suspect the cube was rotated, she will have no evidence by which she can prove it. This pairing of before-and-after questions thus provides a precise criterion by which the presence of a symmetry can be determined.

In physics, the intuitive concept of object symmetry is extended to include any form of change that does not alter the way an object or system behaves. For example, while it is well known that changing where you are located in space should not alter any of the fundamental laws of physics such as conservation of energy or momentum, it is less obvious that this too is an example of a symmetry. In this case the "action" is not rotation of a simple object, but of translation — that is, of moving the entire system, including implicitly the observer — to a new location. As before, the critical question to ask is whether or not the observer, who presumably fell asleep during the move, can tell whether a change in location took place solely by comparing the properties of the system before and after the move. For the fundamental laws such as conservation of energy and momentum, the answer to this question has been determined to be no. Therein lies a truly profound observation about the universe: No matter where one goes within it, certain fundamental laws of classical physics appear to remain exactly the same. Those rules thus are "universal" in the most literal meaning of the word.

This remarkable conclusion from observational physics can be captured precisely and succinctly by using the mathematical concept of operators. In the most general sense, the expression of a classical mechanics symmetry using operators has this form:


 * $$\forall S \in G: F(S(X)) = F(X)$$

While cryptic in appearance, the above equation simply restates what has already been said: A symmetry is an action that leaves no detectable change in an object or system.

In the equation $$S$$ represents one of those undetectable actions on an object or system, such as rotation of a cube or the relocation of a system in space. This is the "operator" that acts (operates) on the original system, such as by rotating an object or moving a system. $$G$$ represents all of the possible actions of that type for which the object or system remains unchanged. For example, for a cube the set $$G$$ includes all twenty-four of the simplest rotations that if done on a perfect cube leave no trace by which an observer can subsequently detect that such an operation or action took place. . Such sets of "no detectable change" operations on an object or system turn out to have interesting and well-defined properties in their own right, and in mathematics are called groups.

In the case of moving through space, the set $$G$$ is very large indeed, since it encompasses every possible point in the universe into which an object or system could be moved! The underlying principle remains exactly the same, however, with $$G$$ simply representing a much larger set of choices than was the case for the 24 invisible rotations of a cube.

Finally, the two occurrences of the letter $$F$$ in the equation represents the two instances of observation of the object or system. The $$F$$ on the right side represents the first observation, taken before the action occurs. The $$F$$ on the left side of the equation represents the second observation that takes place after the action, represented by the operation $$S$$, takes place. The equality sign succinctly summarizes the undetectability of the change: Whatever results the observer obtained the first time, either through direct viewing of the object or system or through detailed experimentation, will turn out to match precisely with the same observations taken after the action, or operation, took place.

A literal translation of the

Consider a classical system with Hamiltonian (energy) function $$H(q,p,t)$$, where $$q$$ are the generalized position coordinates, $$p$$ are the conjugate momenta, and $$t$$ is time. If $$H(q,p)$$ remains unchanged (invariant) when transformed by any element $$S$$ of the group $$G$$ — that is, if for all $$S\in G$$ it remains true that $$H(S(q,p))=H(q,p)$$ — then the elements of $$G$$ are physical operators that map physical states among themselves.

An easy example is given by space translations. The Hamiltonian of a translationally invariant problem does not change under the transformation $$q\to T_a q=q+a$$. Other straightforward symmetry operators are the ones implementing rotations.

If the physical system is described by a function, as in classical field theories, the translation operator is generalized in a straightforward way:

$$f(x) \to T_a f(x)=f(x-a)$$

Notice that the transformation inside the parenthesis should be the inverse of the transformation done on the coordinates.

Concept of generator
If the transformation is infinitesimal, the operator action should be of the form

$$ I + \epsilon A $$

where $$I$$ is the identity operator, $$\epsilon$$ is a small parameter, and $$A$$ will depend on the transformation at hand, and is called a generator of the group. Again, as a simple example, we will derive the generator of the space translations on 1D functions.

As it was stated, $$T_a f(x)=f(x-a)$$. If $$a=\epsilon$$ is infinitesimal, then we may write

$$T_\epsilon f(x)=f(x-\epsilon)\approx f(x) - \epsilon f'(x)$$

This formula may be rewritten as

$$T_\epsilon f(x) = (I-\epsilon D) f(x)$$

where $$D$$ is the generator of the translation group, which happens to be just the derivative operator. Thus, it is said that the generator of translations is the derivative.

The exponential map
The whole group may be recovered, under normal circumstances, from the generators, via the exponential map. In the case of the translations the idea works like this.

The translation for a finite value of $$a$$ may be obtained by repeated application of the infinitesimal translation:

$$T_a f(x) = \lim_{N\to\infty} T_{a/N} \cdots T_{a/N} f(x)$$

with the $$\cdots$$ standing for the application $$N$$ times. If $$N$$ is large, each of the factors may be considered to be infinitesimal:

$$T_a f(x) = \lim_{N\to\infty} (I -(a/N) D)^N f(x)$$

But this limit may be rewritten as an exponential:

$$T_a f(x)= \exp(-aD) f(x)$$

To be convinced of the validity of this formal expression, we may expand the exponential in a power series:

$$T_a f(x) = \left( I - aD + {a^2D^2\over 2!} - {a^3D^3\over 3!} + \cdots \right) f(x)$$

The rhs may be rewritten as

$$f(x) - a f'(x) + {a^2\over 2!} f(x) - {a^3\over 3!} f'(x) + \cdots$$

which is just the Taylor expansion of $$f(x-a)$$, which was our original value for $$T_a f(x)$$.

Operators in quantum mechanics
Once the interest of the operators in classical mechanics has been exposed, it has to be said that it is in quantum mechanics where they reach their full interest. The mathematical description of quantum mechanics is built upon the concept of operator.

Physical pure states in quantum mechanics are unit-norm vectors in a certain vector space (a Hilbert space). Time evolution in this vector space is given by the application of a certain operator, called the evolution operator. Since the norm of the physical state should stay fixed, the evolution operator should be unitary. Any other symmetry, mapping a physical state into another, should keep this restriction.

Any observable, i.e., any quantity which can be measured in a physical experiment, should be associated with a self-adjoint linear operator. The values which may come up as the result of the experiment are the eigenvalues of the operator. The probability of each eigenvalue is related to the projection of the physical state on the subspace related to that eigenvalue.

General mathematical properties of quantum operators
The mathematical properties of physical operators are a topic of great importance in itself. For further information, see C*-algebra and Gelfand-Naimark theorem.