Newton series expansion of bosonic operator functions

We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.We show how series expansions of functions of bosonic number operators are naturally derived from finite-difference calculus. The scheme employs Newton series rather than Taylor series known from differential calculus, and also works in cases where the Taylor expansion fails. For a function of number operators, such an expansion is automatically normal ordered. Applied to the Holstein-Primakoff representation of spins, the scheme yields an exact series expansion with a finite number of terms and, in addition, allows for a systematic expansion of the spin operators that respects the spin commutation relations within a truncated part of the full Hilbert space. Furthermore, the Newton series expansion strongly facilitates the calculation of expectation values with respect to coherent states. As a third example, we show that factorial moments and factorial cumulants arising in the context of photon or electron counting are a natural consequence of Newton series expansions. Finally, we elucidate the connection between normal ordering, Taylor and Newton series by determining a corresponding integral transformation, which is related to the Mellin transform.

arXiv:2008.11139SciPost PhysicsJan 13, 20213 citations

Ordered Expansions in Boson Amplitude Operators

Kevin E. Cahill, Roy J. Glauber

The expansion of operators as ordered power series in the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ 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 $s=+1,0,\ensuremath{-}1$, 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 $s>\frac{1}{2}$ . 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 $f(\ensuremath{\alpha})$ associated with a given operator $F$ is the one which results when the operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for $F$ are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ 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 expansion of operators as ordered power series in the annihilation and creation operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ 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 $s=+1,0,\ensuremath{-}1$, 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 $s>\frac{1}{2}$. 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 $f(\ensuremath{\alpha})$ associated with a given operator $F$ is the one which results when the operators $a$ and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for $F$ are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ 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 $s$.

Physical ReviewJan 25, 19691,117 citations

Resummation of the Holstein-Primakoff expansion and differential equation approach to operator square roots

Michael Rudolf Vogl, Pontus Laurell, Hao Zhang +2 more

Operator square roots are ubiquitous in theoretical physics.They appear, for example, in the Holstein-Primakoff representation of spin operators and in the Klein-Gordon equation.Often the use of a perturbative expansion is the only recourse when dealing with them.In this paper, we show that under certain conditions, differential equations can be derived which can be used to find perturbatively inaccessible approximations to operator square roots.Specifically, for the number operator n = a † a we show that the square root √ n near n = 0 can be approximated by a polynomial in n.This result is unexpected because a Taylor expansion fails.A polynomial expression in n is possible because n is an operator, and its constituents a and a † have a non trivial commutator [a, a † ] = 1 and do not behave as scalars.We apply our approach to the zero-mass Klein-Gordon Hamiltonian in a constant magnetic field and, as a main application, the Holstein-Primakoff representation of spin operators, where we are able to find new expressions that are polynomial in bosonic operators.We prove that these new expressions exactly reproduce spin operators.Our expressions are manifestly Hermitian, which offers an advantage over other methods, such as the Dyson-Maleev representation.Operator square roots are ubiquitous in theoretical physics.They appear, for example, in the Holstein-Primakoff representation of spin operators and in the Klein-Gordon equation.Often the use of a perturbative expansion is the only recourse when dealing with them.In this paper, we show that under certain conditions, differential equations can be derived which can be used to find perturbatively inaccessible approximations to operator square roots.Specifically, for the number operator n = a † a we show that the square root √ n near n = 0 can be approximated by a polynomial in n.This result is unexpected because a Taylor expansion fails.A polynomial expression in n is possible because n is an operator, and its constituents a and a † have a non trivial commutator [a, a † ] = 1 and do not behave as scalars.We apply our approach to the zero-mass Klein-Gordon Hamiltonian in a constant magnetic field and, as a main application, the Holstein-Primakoff representation of spin operators, where we are able to find new expressions that are polynomial in bosonic operators.We prove that these new expressions exactly reproduce spin operators.Our expressions are manifestly Hermitian, which offers an advantage over other methods, such as the Dyson-Maleev representation.

arXiv:2006.06871Physical Review ResearchNov 17, 202035 citations

A note on normal ordering of degenerate integral powers of number operator

Taekyun Kim, Dae San Kim, Kyo Shin Hwang

This study derives the normal ordering expansion of degenerate integral powers of the number operator, (a†a)_{n,λ}, using recurrence relations for the coefficients and an operator action on number states. Here a† and a are respectively the boson creation and annihilation operators. We also determine the inverse of this normal ordering expansion. By analyzing diagonal coherent state elements of the degenerate integral powers of the number operator, we establish a combinatorial identity which yields a Dobinski-like formula for the degenerate Bell numbers at a specific value, connecting degenerate quantum operator calculus with combinatorics.This study derives the normal ordering expansion of degenerate integral powers of the number operator, (a†a)_{n,λ}, using recurrence relations for the coefficients and an operator action on number states. Here a† and a are respectively the boson creation and annihilation operators. We also determine the inverse of this normal ordering expansion. By analyzing diagonal coherent state elements of the degenerate integral powers of the number operator, we establish a combinatorial identity which yields a Dobinski-like formula for the degenerate Bell numbers at a specific value, connecting degenerate quantum operator calculus with combinatorics.

Notes on Number Theory and Discrete MathematicsNov 5, 20250 citations

Convergence of Boson Expansion Theory

Motoyuki Sato, Kosuke Takada

When the Holstein-Primakoff (Hermitian)-type boson expansion theory is applied for nuclear structure calculations, truncation at the fourth-order expansion of the boson Hamiltonian is often used. The Dyson-type boson mapping has no difficulty concerning boson expansion, since the Dyson boson Hamiltonian is written in up to sixth-order boson operators, and we do not have to make any truncation of the boson expansion. In the present work, we propose a new clear-cut method to obtain the normal-ordered Holstein-Primakoff boson expansion. Using this method, we test whether or not the truncation of the boson expansion is good approximation in various cases. We show that truncation at the fourth-order expansion of the normal-ordered Holstein-Primakoff boson Hamiltonian is a rather good approximation, but it is not sufficient for detailed analyses.When the Holstein-Primakoff (Hermitian)-type boson expansion theory is applied for nuclear structure calculations, truncation at the fourth-order expansion of the boson Hamiltonian is often used. The Dyson-type boson mapping has no difficulty concerning boson expansion, since the Dyson boson Hamiltonian is written in up to sixth-order boson operators, and we do not have to make any truncation of the boson expansion. In the present work, we propose a new clear-cut method to obtain the normal-ordered Holstein-Primakoff boson expansion. Using this method, we test whether or not the truncation of the boson expansion is good approximation in various cases. We show that truncation at the fourth-order expansion of the normal-ordered Holstein-Primakoff boson Hamiltonian is a rather good approximation, but it is not sufficient for detailed analyses.

Progress of Theoretical PhysicsSep 1, 19981 citation

General expansion of natural power of linear combination of Bosonic operators in normal order

Deepak Deepak, Arpita Chatterjee

In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation ( $a^\dagger$ ) and annihilation ( $a$ ) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator $(a^\dagger x+ay)^n$ with $n$ as the exponent and $x,\,y$ are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation ($a^\dagger$) and annihilation ($a$) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator $(a^\dagger x+ay)^n$ with $n$ as the exponent and $x,\,y$ are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.

arXiv:2305.18113May 29, 20230 citations

The expansion of operators as ordered power series in the annihilation and creation operators

a

and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ 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 $s=+1,0,\ensuremath{-}1$, 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>\frac{1}{2}

. 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 $f(\ensuremath{\alpha})$ associated with a given operator

F

is the one which results when the operators

a

and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ occurring in the ordered power series for

F

are replaced by their complex eigenvalues $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{*}$. The correspondence which is realized for symmetric ordering is the Weyl correspondence. The operators associated by each correspondence with the set of $\ensuremath{\delta}$ 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

In quantum mechanics, bosonic operators are mathematical objects that are used to represent the creation (

a^\dagger

) and annihilation (

a

) of bosonic particles. The natural power of a linear combination of bosonic operators represents an operator

(a^\dagger x+ay)^n

with

n

as the exponent and

x,\,y

are the variables free from bosonic operators. The normal ordering of these operators is a mathematical technique that arranges the operators so that all the creation operators are to the left of the annihilation operators, reducing the number of terms in the expression. In this paper, we present a general expansion of the natural power of a linear combination of bosonic operators in normal order. We show that the expansion can be expressed in terms of binomial coefficients and the product of the normal-ordered operators using the direct method and than prove it using the fundamental principle of mathematical induction. We also derive a formula for the coefficients of the expansion in terms of the number of bosons and the commutation relation between the creation and annihilation operators. Our results have important applications in the study of many-body systems in quantum mechanics, such as in the calculation of correlation functions and the evaluation of the partition function. The general expansion presented in this paper provides a powerful tool for analyzing and understanding the behavior of bosonic systems, and can be applied to a wide range of physical problems.