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Normal Ordering and Abnormal Nonsense
Abstract
The technique of the normal ordering of non-commuting operators is an important tool in the solution of problems involving creation and annihilation operators in quantum physics, such as in many-body theory or quantum optics. We point out the inconsistencies in previous definitions of the two standard normal ordering procedures for such operators, and show how consistent definitions may be made.
... This in turn paved the way for finding generating functions encoding urn histories as indicated in Eqs. (22) and (23) -in fact the tasks have been proved to be equivalent. ...
... Therefore, the double dot operation is not of a direct use in calculus. See Ref.[22] for discussion. ...
Non-commutativity is ubiquitous in mathematical modeling of reality and in many cases same algebraic structures are implemented in different situations. Here we consider the canonical commutation relation of quantum theory and discuss a simple urn model of the latter. It is shown that enumeration of urn histories provides a faithful realization of the Heisenberg-Weyl algebra. Drawing on this analogy we demonstrate how the operator normal forms facilitate counting of histories via generating functions, which in turn yields an intuitive combinatorial picture of the ordering procedure itself.
... Therefore, the double dot operation is not of a direct use in calculus. See Ref. [22] for discussion. X k D l which withdraw l balls all at once and then replace them with k new ones. ...
Non-commutativity is ubiquitous in mathematical modeling of reality and in many cases same algebraic structures are implemented in different situations. Here we consider the canonical commutation relation of quantum theory and discuss a simple urn model of the latter. It is shown that enumeration of urn histories provides a faithful realization of the Heisenberg–Weyl algebra. Drawing on this analogy we demonstrate how the operator normal forms facilitate counting of histories via generating functions, which in turn yields an intuitive combinatorial picture of the ordering procedure itself.
Reduced density matrix cumulants play key roles in the theory of both reduced density matrices and multiconfigurational normal ordering. We present a new, simpler generating function for reduced density matrix cumulants that is formally identical to equating the coupled cluster and configuration interaction ans{\"a}tze. This is shown to be a general mechanism to convert between a multiplicatively separable quantity and an additively separable quantity, as defined by a set of axioms. It is shown that both the cumulants of probability theory and reduced density matrices are entirely combinatorial constructions, where the differences can be associated to changes in the notion of ``multiplicative separability'' for expectation values of random variables compared to reduced density matrices. We compare our generating function to that of previous works and criticize previous claims of probabilistic significance of the reduced density matrix cumulants. Finally, we present a simple proof of the Generalized Normal Ordering formalism to explore the role of reduced density matrix cumulants therein. While the formalism can be used without cumulants, the combinatorial structure of expressing RDMs in terms of cumulants is the same combinatorial structure on cumulants that allows for a simple extended generalized Wick's Theorem.
Reduced density matrix cumulants play key roles in the theory of both reduced density matrices and multiconfigurational normal ordering, but the underlying formalism has remained mysterious. We present a new, simpler generating function for reduced density matrix cumulants that is formally identical to equating the coupled cluster and configuration interaction ansätze. This is shown to be a general mechanism to convert between a multiplicatively separable quantity and an additively separable quantity, as defined by a set of axioms. It is shown that both the cumulants of probability theory and reduced density matrices are entirely combinatorial constructions, where the differences can be associated to changes in the notion of "multiplicative separability'' for expectation values of random variables compared to reduced density matrices. We compare our generating function to that of previous works and criticize previous claims of probabilistic significance of the reduced density matrix cumulants. Finally, we present the simplest proof to date of the Generalized Normal Ordering formalism to explore the role of reduced density matrix cumulants therein.
We introduce the general method of converting a given operator function into its s-ordered form. We state and prove a theorem representing the fact that any ordered expansion of some operator function might be considered as the combinatorial problem of counting the number of contractions. This will also unify the two essentially distinct notions of 'taking an operator into some ordered form' and 'reordering, or formally, ordering an operator'. In this way, we reduce the general ordering problem into a purely combinatorial one. Finally, we show the application of the theorem through two generic examples from both quantum optics and field theory.
We discuss a general combinatorial framework for operator ordering problems by applying it to the normal ordering of the powers and exponential of the boson number operator. The solution of the problem is given in terms of Bell and Stirling numbers enumerating partitions of a set. This framework reveals several inherent relations between ordering problems and combinatorial objects, and displays the analytical background to Wick's theorem. The methodology can be straightforwardly generalized from the simple example we discuss to a wide class of operators.
The expansion of operators as ordered power series in the annihilation and creation operators a and a† is examined. It is found that normally ordered power series exist and converge quite generally, but that for the case of antinormal ordering the required c-number coefficients are infinite for important classes of operators. A parametric ordering convention is introduced according to which normal, symmetric, and antinormal ordering correspond to the values s=+1,0,-1, respectively, of an order parameter s. In terms of this convention it is shown that for bounded operators the coefficients are finite when s>0, and the series are convergent when s>12. For each value of the order parameter s, a correspondence between operators and c-number functions is defined. Each correspondence is one-to-one and has the property that the function f(alpha) associated with a given operator F is the one which results when the operators a and a† occurring in the ordered power series for F are replaced by their complex eigenvalues alpha and alpha*. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of delta functions on the complex plane are discussed in detail. They are shown to furnish, for each ordering, an operator basis for an integral representation for arbitrary operators. The weight functions in these representations are simply the functions that correspond to the operators being expanded. The representation distinguished by antinormal ordering expresses operators as integrals of projection operators upon the coherent states, which is the form taken by the P representation for the particular case of the density operator. The properties of the full set of representations are discussed and are shown to vary markedly with the order parameter s.
The advent of lasers in the 1960s led to the development of many new
fields in optical physics. This book is a systematic treatment of one of
these fields--the broad area that deals with the coherence and
fluctuation of light. The authors begin with a review of probability
theory and random processes, and follow this with a thorough discussion
of optical coherence theory within the framework of classical optics.
They next treat the theory of photoelectric detection of light and
photoelectric correlation. They then discuss in some detail quantum
systems and effects. The book closes with two chapters devoted to laser
theory and one on the quantum theory of nonlinear optics. The sound
introduction to coherence theory and the quantum nature of light and the
chapter-end exercises will appeal to graduate students and newcomers to
the field. Researchers will find much of interest in the new results on
coherence-induced spectral line shifts, nonclassical states of light,
higher-order squeezing, and quantum effects of down-conversion. Written
by two of the world's most highly regarded optical physicists, this book
is required reading of all physicists and engineers working in optics.
An alteration in the notation used to indicate the order of operation of noncommuting quantities is suggested. Instead of the order being defined by the position on the paper, an ordering subscript is introduced so that AsBs′ means AB or BA depending on whether s exceeds s′ or vice versa. Then As can be handled as though it were an ordinary numerical function of s. An increase in ease of manipulating some operator expressions results. Connection to the theory of functionals is discussed in an appendix. Illustrative applications to quantum mechanics are made. In quantum electrodynamics it permits a simple formal understanding of the interrelation of the various present day theoretical formulations.
The operator expression of the Dirac equation is related to the author's previous description of positrons. An attempt is made to interpret the operator ordering parameter in this case as a fifth coordinate variable in an extended Dirac equation. Fock's parametrization, discussed in an appendix, seems to be easier to interpret.
In the last section a summary of the numerical constants appearing in formulas for transition probabilities is given.
Note on the representation of finite continuous groups by means of linear substitutions
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