Applications of Supersymmetric Polynomials in Statistical Quantum Physics

We propose a correspondence between partition functions of ideal gases consisting of both bosons and fermions and algebraic bases of supersymmetric polynomials on the Banach space of absolutely summable two-sides sequences ℓ1(Z0). Such an approach allows us to interpret some combinatorial identities for supersymmetric polynomials from a physical point of view. We consider a relation of equivalence on ℓ1(Z0) induced by the supersymmetric polynomials, and semiring algebraic structures on the quotient set with respect to this relation. The quotient set is a natural model for the set of energy levels of a quantum system. We introduce two different topological semiring structures on this set and discuss their possible physical interpretations.

Keywords:

Subject: Physical Sciences - Quantum Science and Technology

MSC: 46G25; 46N50

1. Introduction

Symmetric polynomials variables and relations between bases of the algebra of symmetric polynomials are widely used in Algebra, Combinatorics (see [1]), and in particular, in Statistical Quantum Mechanics. In [2,3] Schmidt and Schnack proposed some correspondence between relations in the algebra of symmetric polynomials and partition functions of bosons and fermions. Under this correspondence, one basis of symmetric polynomials is responsible for bosons and another for fermions. Such an approach was applied and developed for different cases by many authors (see e.g. [4,5,6,7,8]). On the other hand, recently some new results for algebras of symmetric analytic functions on infinite-dimensional Banach spaces were obtained [9,10,11,12,13,14]. The infinite number of variables of the underlying space allows us to introduce some interesting algebraic operations on the spectra of such algebras that may have a physical meaning. In addition, in the infinite-dimensional case, we can consider the behavior of the ideal gas “at infinity” if, for example, the number of particles grows to infinity while the total energy of the system is bounded.

In [15,16,17] were considered supersymmetric polynomials and analytic functions on abstract Banach spaces. Supersymmetric polynomials of several variables were studied in [18,19,20]. It seems to be that some bases of supersymmetric polynomials give us a tool for the investigation of a quantum ideal gas consisting of both bosons and fermions. Also, supersymmetric polynomials define a relation of equivalence on the underlying vector space and the quotient set with respect to this relation looks like the most natural model for the set of energy levels of a given quantum system. Such a set admits some algebraic semiring structures, related, in particular, to tropical (idempotent) mathematics.

In this paper, we discuss relations between algebras of supersymmetric polynomials on Banach spaces and partition functions of bosons and fermions and consider some new algebraic structures on the set of energy levels of corresponding quantum systems.

In Section 2 we gather basically known information about algebraic bases of symmetric polynomials on the Banach space

ℓ_{1}

and their relations to partition functions of ideal quantum gases. In Section 3 we consider algebraic bases of supersymmetric polynomials and discuss their relations to partition functions of ideal gases consisting simultaneously of bosons and fermions. In Section 4 we construct two different semiring structures on set of energy levels. The first one is related to algebraic operations that were introduced in [17] for a more general case. The second is related to the idempotent operation max and looks like an infinite-dimensional generalization of the tropical semiring

R \cup + \infty

(c.f. [21]).

General information on polynomials and analytic functions on abstract Banach spaces can be found in [22,23]. The idempotent analysis and tropical semirings are considered in [24,25].

2. Preliminaries results on symmetric polynomials and partition functions

2.1. Symmetric polynomials

Let

N

be the set of all positive integers, and

ℓ_{1}

be the Banach space of all absolutely summing complex sequences

x = (x_{1}, \dots, x_{n}, \dots)

with norm

∥ x ∥ = \sum_{n = 1}^{\infty} | x_{n} | .

A function f on

ℓ_{1}

is called symmetric if

f ((x_{σ (1)}, x_{σ (2)}, \dots)) = f ((x_{1}, x_{2}, \dots))

for every

(x_{1}, x_{2}, \dots) \in ℓ_{1}

and every bijection

σ : N \to N .

Let us define following symmetric polynomials on

ℓ_{1} .

Let the polynomial

F_{n}

be defined by

F_{n} ((x_{1}, x_{2}, \dots)) = \sum_{i = 1}^{\infty} x_{i}^{n},

(1)

where

n \in N .

Polynomials

F_{n}

are called power sum symmetric polynomials. Let us define polynomials

B_{n}

B_{n} ((x_{1}, x_{2}, \dots)) = \sum_{i_{1} \leq \dots \leq i_{n}} x_{i_{1}} \dots x_{i_{n}},

(2)

where

n \in N .

Polynomials

B_{n}

are called complete symmetric polynomials. Let the polynomial

G_{n}

be defined by

G_{n} ((x_{1}, x_{2}, \dots)) = \sum_{i_{1} < \dots < i_{n}} x_{i_{1}} \dots x_{i_{n}},

(3)

where

n \in N .

Polynomials

G_{n}

are called elementary symmetric polynomials.

Definition 1.

A linear combination of finite products of powers (zero powers are also allowed) of elements of an algebra is called an algebraic combination of these elements.

A subset of an algebra is called algebraically independent if zero element of the algebra cannot be represented as a nontrivial algebraic combination of elements of this subset.

An algebraically independent subset of an algebra is called an algebraic basis of this algebra if every element of the algebra can be represented as an algebraic combination of elements of the subset. Due to the algebraic independence every such a representation is unique.

Let

P_{s} (ℓ_{1})

denotes the algebra of all continuous symmetric complex-valued polynomials on

ℓ_{1} .

Every set of polynomials

{F_{n} : n \in N},

{B_{n} : n \in N}

and

{G_{n} : n \in N}

is an algebraic basis in

P_{s} (ℓ_{1})

(see e.g. [9,13]). There are so-called Newton recurrent formulas connecting different algebraic bases:

m G_{m} = \sum_{k = 1}^{m} {(- 1)}^{k - 1} G_{m - k} F_{k}, m \in N,

(4)

m B_{m} = \sum_{k = 1}^{m} B_{m - k} F_{k}, m \in N,

(5)

G_{m} = \sum_{k = 1}^{m} {(- 1)}^{k - 1} G_{m - k} B_{k}, m \in N,

(6)

and

B_{m} = \sum_{k = 1}^{m} {(- 1)}^{k - 1} B_{m - k} G_{k}, m \in N .

(7)

Let

B (x) (t)

and

G (x) (t)

be the so-called generating functions for polynomials

B_{n}

and

G_{n}

respectively, defined as the following formal series

B (x) (t) = \sum_{n = 0}^{\infty} t^{n} B_{n} (x), B_{0} = 1

(8)

and

G (x) (t) = \sum_{n = 0}^{\infty} t^{n} G_{n} (x), G_{0} = 1 .

(9)

The following relations are well-known ([1], p. 3)

G (x) (t) = exp (- \sum_{n = 1}^{\infty} t^{n} \frac{F_{n} (- x)}{n}) a n d B (x) (t) = exp (\sum_{n = 1}^{\infty} t^{n} \frac{F_{n} (x)}{n}),

(10)

and they immediately imply that

G (x) (t) B (- x) (t) = 1 .

(11)

Here the equality holds for every

x \in ℓ_{1}

and for every t in the common domain of convergence. Note that

G (x) (t)

is a well-defined analytic function of

x \in ℓ_{1}

for every fixed

t \in C

and a function of exponential type of t for every fixed x [26].

2.2. Partition functions

The canonical partition function plays a fundamental role in statistical mechanics since most thermodynamic functions can be derived from it [3]. It is defined by

Z_{N} (β) = Tr exp (- β H),

(12)

where H denotes the Hamiltonian of the system, N is the number of particles and

β = \frac{1}{k_{B} T}

denotes the inverse temperature (

k_{B}

is the Boltzmann constant, T is the temperature). In other words, H is a self adjoint operator such that

exp (- β H)

is a trace class operator for

β \in R .

The grand canonical partition function is defined by

Z (z, β) = \sum_{N = 0}^{\infty} Z_{N} (β) z^{N},

(13)

where the variable z is physically interpreted as the fugacity of the system, i.e.,

z = exp (μ / (k_{B} T))

(

μ

is the chemical potential). It describes the system in which the number of particles can be changed. The physical interpretation implies that z must be non-negative.

Note that the partition function completely defines all possible states of the system. Also it can be used for deriving the possibilities of states.

Consider the ideal gas consisting of noninteracting identical particles (bosons or fermions). In this case the Hamiltonian H is the sum of N identical single-particle Hamiltonians:

H = \sum_{n = 1}^{N} h_{n} .

Let

E_{i}

be single-particle energy eigenvalues. In [27] it is shown that

Z_{N} (β) = B_{N} ((x_{1}, x_{2}, \dots))

(14)

for the system of bosons and

Z_{N} (β) = G_{N} ((x_{1}, x_{2}, \dots))

(15)

for the system of fermions, where

B_{N}

is defined by (2),

G_{N}

is defined by (3) and

x_{i} = exp (- β E_{i}) .

(16)

Note that

Z_{N}

is a symmetric function between energy levels, not between particles.

By (8), (9), (13), (14) and (15), the grand canonical partition function can be represented in the form

Z (z, β) = B ((x_{1}, x_{2}, \dots)) (z)

for bosons and

Z (z, β) = G ((x_{1}, x_{2}, \dots)) (z)

for fermions, where

x_{i}

are defined by (16). In addition, according to [2] the coordinates

(x_{1}, x_{2}, \dots)

x \in ℓ_{1}

correspond to abstract energy levels of the system, a monomial

x_{1}^{n_{1}} \dots x_{m}^{n_{m}},

n_{1} + \dots + n_{m} = N

in a partition function corresponds to possible occupation of levels

x_{1}, \dots, x_{m}

by N particles. Also, there exists so-called fundamental symmetry

ω

P_{s} (ℓ_{1})

which is defined as an algebra homomorphism from

P_{s} (ℓ_{1})

to itself such that

ω (F_{n}) = {(- 1)}^{n - 1} F_{n}

n \in N .

In other words, for every

n,

(ω (F_{n})) (x) = - F_{n} (x) .

Note that

ω

is an involution in the sense that

ω^{2}

is the unity operator. It is known that

ω G_{n} = B_{n}

and

ω B_{n} = G_{n}

for every

n \in N

[1]. In [2] was observed that Newton’s identity (4) corresponds to Landsberg’s identity in physics [28] and equation (11) is related to the Bose-Fermi symmetry.

2.3. Note about the Banach space $ℓ_{1}$

As we mentioned above,

exp (- β H)

is a trace class operator and so, its eigenvalues

x_{i} = exp (- β E_{i})

are summable, that is,

x = (x_{1}, x_{2}, \dots) \in ℓ_{1} .

On the other hand, in [2] it was observed that for the case

n = \infty,

the evaluations

G

and

B

leads to corresponding grand canonical partition functions only if these series converge. Since all

x_{i} \geq 0,

the vector x must be in

ℓ_{1} .

Thus, the space of absolutely summable sequences

ℓ_{1}

is a most natural domain for vectors

x = (x_{1}, x_{2}, \dots)

and

P_{s} (ℓ_{1})

is a most natural algebra of symmetric polynomials for

n = \infty .

However, it is possible to consider symmetric polynomials in the general case

ℓ_{p},

1 \leq p < \infty

and even for the case of “continual” number of variables if

x \in L_{p},

1 \leq p \leq \infty

(see [13,29,30,31] and references therein).

Note that in [32] were considered some relations between a trace class operator A and the (infinite-dimensional) Fredholm determinant

det (I - A),

where I is the identity operator. In particular, if A is self-adjoint with eigenvalues

x_{i},

then

det (I - t A) = G (x) (t) and {(det (I - A))}^{- 1} = B (- x) (t) .

Applications of determinants of the form

det (I - A)

to partition functions can be found in [33].

3. Supersymmetric polynomials and partition functions for mixed systems of bosons and fermions

Let

Z

be the set of all integers and

Z_{0} = Z ∖ {0} .

We denote by

ℓ_{1} (Z_{0})

the Banach space of all absolutely summing complex sequences indexed by elements of

Z_{0}

(two-sides sequences). Every element of

ℓ_{1} (Z_{0})

can be represented in the form

(y | x) = (\dots, y_{2}, y_{1} | x_{1}, x_{2}, \dots)

with

∥ (y | x) ∥ = \sum_{i = 1}^{\infty} (| x_{i} | + | y_{i} |),

where

x = (x_{1}, x_{2}, \dots)

and

y = (y_{1}, y_{2}, \dots)

belong to

ℓ_{1} .

For every

n \in N

we define the polynomials

T_{n},

n \in N

ℓ_{1} (Z_{0})

T_{n} ((y | x)) = F_{n} (x) - F_{n} (y),

where

F_{n}

is defined by (1).

A polynomial on

ℓ_{1} (Z_{0})

is called supersymmetric (see [17]) if it can be represented as an algebraic combination of elements of the set

{T_{n} : n \in N} .

Let us denote

P_{s u p} (ℓ_{1} (Z_{0}))

the algebra of all supersymmetric polynomials on

ℓ_{1} (Z_{0}) .

Note that the set

{T_{n} : n \in N}

is an algebraic basis of the algebra

P_{s u p} (ℓ_{1} (Z_{0})) .

Let us define another important supersymmetric polynomials on

ℓ_{1} (Z_{0})

which also form an algebraic basis of the algebra

P_{s u p} (ℓ_{1} (Z_{0})) .

For

n \in N

let

W_{n} : ℓ_{1} (Z_{0}) \to C

be defined by

W_{n} ((y | x)) = \sum_{k = 0}^{n} G_{k} (x) B_{n - k} (- y) .

(17)

Note that polynomials

W_{n}

can be obtained if we substitute in Newton’s formula (4) polynomials

T_{n}

instead of

F_{n}

[17]. In other words,

m W_{m} ((y | x)) = \sum_{k = 1}^{m} {(- 1)}^{k - 1} W_{m - k} ((y | x)) T_{k} ((y | x)), m \in N .

(18)

From (18), in particular, it follows that all polynomials

W_{n}

are supersymmetric and form an algebraic basis in

P_{s u p} (ℓ_{1} (Z_{0})) .

Let

W ((y | x)) (t)

be the formal series

W ((y | x)) (t) = \sum_{n = 0}^{\infty} t^{n} W_{n} ((y | x)), W_{0} = 1,

(19)

that is,

W

is the generating function for polynomials

W_{n} .

By [17],

W ((y | x)) (t) = \frac{G (x) (t)}{G (y) (t)},

(20)

where the equality is true on the common domain of convergence.

Consider a mixed system of bosons and fermions. In [27] it is shown that the partition function for the system, where the total number N of bosons and fermions is fixed, can be represented in the form

Z_{N} (β) = \sum_{k = 0}^{N} G_{k} ((x_{1}^{(F)}, x_{2}^{(F)}, \dots)) B_{N - k} ((x_{1}^{(B)}, x_{2}^{(B)}, \dots)),

(21)

where

x_{i}^{(F)} = exp (- β E_{i}^{(F)}) and x_{i}^{(B)} = exp (- β E_{i}^{(B)}),

E_{i}^{(F)}

and

E_{i}^{(B)}

are single-particle energies of fermions and bosons resp.

Let

{\tilde{W}}_{n} : ℓ_{1} (Z_{0}) \to C

be defined by

{\tilde{W}}_{n} ((y | x)) = W_{n} ((- x | - y)),

(22)

where

W_{n}

is defined by (17). By (17) and (22),

{\tilde{W}}_{n} ((y | x)) = \sum_{k = 0}^{n} G_{k} (- y) B_{n - k} (x) .

(23)

By (21) and (23),

Z_{N} (β) = {\tilde{W}}_{N} ((\tilde{y} | \tilde{x})),

(24)

where

\tilde{y} = (- x_{1}^{(F)}, - x_{2}^{(F)}, \dots)

(25)

and

\tilde{x} = (x_{1}^{(B)}, x_{2}^{(B)}, \dots) .

(26)

If sequences are finite, we complete them with an infinite number of zeros. Note that the equality (24) makes sense only if

\tilde{x}

and

\tilde{y}

belong to

ℓ_{1} .

Otherwise we only can consider (24) as formal equality.

Let us consider the grand canonical partition function. By (13) and (24),

Z (z, β) = \sum_{N = 0}^{\infty} z^{N} {\tilde{W}}_{N} ((\tilde{y} | \tilde{x})), {\tilde{W}}_{0} = 1 .

(27)

For

(y | x) \in ℓ_{1} (Z_{0})

and

t \in C

let

\tilde{W} ((y | x)) (t)

be the formal series

\tilde{W} ((y | x)) (t) = \sum_{n = 0}^{\infty} t^{n} {\tilde{W}}_{n} ((y | x)) .

(28)

Evidently,

Z (z, β) = \tilde{W} ((\tilde{y} | \tilde{x})) (z) .

(29)

On the other hand, by (28), (22), (19) and (20),

\begin{matrix} \tilde{W} ((y | x)) (t) & = \sum_{n = 0}^{\infty} t^{n} {\tilde{W}}_{n} ((- x | - y)) \\ = W ((- x | - y)) (t) \\ = \frac{G (- y) (t)}{G (- x) (t)} = \frac{B (x)}{B (y)} . \end{matrix}

(30)

So, by (29), (30) and (11)

Z (z, β) = \frac{G (- \tilde{y}) (t)}{G (- \tilde{x}) (t)} = \frac{B (\tilde{x})}{B (\tilde{y})},

where

\tilde{y}

and

\tilde{x}

are defined by (25) and (26) resp.

Thus, we have represented the grand canonical partition function of the mixed system of bosons and fermions via the generating functions

G

and

B

for elementary symmetric polynomials.

Let us observe that if we apply the transformation

(y | x) \mapsto (- x | - y)

T_{n}

for the case

y = 0,

we will obtain

F_{n} (x) = T_{n} ((0 | x)) \mapsto T_{n} {((- x | 0) = (- 1)}^{n - 1} T_{n} ((0 | x)) = {(- 1)}^{n - 1} F_{n} (x) = (ω (F_{n})) (x) .

In other words, the involution

ω

P_{s} (ℓ_{1})

can be extended to

P_{s u p} (ℓ_{1} (Z_{0}))

setting

(ω (P)) ((y | x)) = P ((- x | - y)) .

In particular,

ω (W_{n}) = {\tilde{W}}_{n} .

Applying the homomorphism

ω

to (18), we obtain

m {\tilde{W}}_{m} ((y | x)) = \sum_{k = 1}^{m} {\tilde{W}}_{m - k} ((y | x)) T_{k} ((y | x)), m \in N,

that is,

{\tilde{W}}_{n}

can be obtained if we substitute

T_{n}

instead of

F_{n}

to the Newton formula (5) and so, we have another representation for

{\tilde{W}}_{n}

which can be interpreted as another realization of Landsberg’s identity. In addition, from (6), (7) we can get

{\tilde{W}}_{m} = \sum_{k = 1}^{m} {(- 1)}^{k - 1} {\tilde{W}}_{m - k} W_{k}, m \in N .

4. Semiring structures on the set of variables

4.1. The ring $M_{0}$

First we consider a dense linear subspace

c_{00}

ℓ_{1} .

Let

c_{00}

be the vector space of all eventually zero sequences of complex numbers. Let

c_{00} (Z_{0})

be the subspace of

ℓ_{1} (Z_{0})

consisting of all

(y | x) \in ℓ_{1} (Z_{0})

such that

x, y \in c_{00} .

To shorten the notation we will write elements of

c_{00}

(x_{1}, \dots, x_{n})

instead of

(x_{1}, \dots, x_{n}, 0, \dots) .

Correspondingly, we will write elements of

c_{00} (Z_{0})

(y_{1}, \dots, y_{m} | x_{1}, \dots, x_{n}) .

Let us define the following equivalence relation on

c_{00} (Z_{0}) .

For

a, b \in c_{00} (Z_{0})

let

a \sim b

if and only if

T_{n} (a) = T_{n} (b)

for every

n \in N .

Let

M_{0} = c_{00} (Z_{0}) / \sim .

Note that we have two types of equivalent elements:

(y_{1}, \dots, y_{m} | x_{1}, \dots, x_{n}) \sim (y_{τ (1)}, \dots, y_{τ (m)} | x_{σ (1)}, \dots, x_{σ (n)}),

where

τ

and

σ

are permutations on sets

{1, \dots, m}

and

{1, \dots, n}

resp., and

(y_{1}, \dots, y_{m}, c | c, x_{1}, \dots, x_{n}) \sim (y_{1}, \dots, y_{m} | x_{1}, \dots, x_{n}) .

Consequently, every element of

M_{0}

has the representative

(y | x),

where

x, y \in c_{00},

such that multisets of nonzero elements of x and y are disjoint. On the other hand, every pair of disjoint finite multisets of nonzero complex numbers define some element of

M_{0} .

So, we have the bijection between

M_{0}

and the set of all pairs of disjoint finite multisets of nonzero complex numbers. Let us define ring operations on

M_{0} .

First we define some auxiliary operations on

c_{00} .

Let

(x_{1}, \dots, x_{n}) • (x_{1}^{'}, \dots, x_{m}^{'}) = (x_{1}, \dots, x_{n}, x_{1}^{'}, \dots, x_{m}^{'})

and

\begin{matrix} (x_{1}, \dots, x_{n}) ⋄ (x_{1}^{'}, \dots, x_{m}^{'}) \\ = (x_{1} x_{1}^{'}, x_{1} x_{2}^{'}, \dots, x_{1} x_{m}^{'}, x_{2} x_{1}^{'}, x_{2} x_{2}^{'}, \dots, x_{2} x_{m}^{'}, \dots, x_{n} x_{1}^{'}, x_{n} x_{2}^{'}, \dots, x_{n} x_{m}^{'}) \end{matrix}

for

(x_{1}, \dots, x_{n}), (x_{1}^{'}, \dots, x_{m}^{'}) \in c_{00} .

Let

[z] + [z^{'}] = [(y • y^{'} | x • x^{'})]

and

[z] [z^{'}] = [((y ⋄ x^{'}) • (x ⋄ y^{'}) | (y ⋄ y^{'}) • (x ⋄ x^{'}))]

for

z = (y | x), z^{'} = (y^{'} | x^{'}) \in c_{00} (Z_{0}),

where

x, y, x^{'}, y^{'} \in c_{00} .

By [17]

M_{0}

with these operations is a ring, where

- [(y | x)] = [(x | y)] .

Note that

M_{0}

is not a linear space, so it is not an algebra [17].

Let

a \sim b .

Since

T_{n} (a) = T_{n} (b)

for every

n \in N,

it follows that

f (a) = f (b)

for every supersymmetric function

f .

That is, the value of a supersymmetric function does not depend on the choice of a representative of a class. So, we can set

f ([a]) = f (a)

for a supersymmetric function f and for

[a] \in M_{0} .

Let us consider how our ring operations interplay with the algebraic basis

T_{n}

and the partition function

\tilde{W} (z) (t) .

By [17],

T_{n} ([z] [z^{'}]) = T_{n} ([z]) T_{n} ([z^{'}]) a n d T_{n} ([z] + [z^{'}]) = T_{n} ([z]) + T_{n} ([z^{'}])

(31)

for every

n \in N

and

[z], [z^{'}] \in M_{0} .

In other words, each

T_{n}

is a ring homomorphism from

M_{0}

C .

Also, it is easy to check (c.f. [17]) that

W ([z] + [z^{'}]) (t) = W ([z]) (t) W ([z^{'}]) (t)

and

\tilde{W} ([z] + [z^{'}]) (t) = \tilde{W} ([z]) (t) \tilde{W} ([z^{'}]) (t) .

The following example may be interesting for evaluating grand canonical partition functions “at infinity”.

Example 1.

Let λ and μ be positive numbers. Set

z^{(n)} = (\underset{n}{\underset{︸}{- \frac{μ}{n}, \dots, - \frac{μ}{n}}} | \underset{n}{\underset{︸}{\frac{λ}{n}, \dots, \frac{λ}{n}}}) .

Taking into account [17] and relations between

W

and

\tilde{W},

we can see that if

n \to \infty,

then both

W (z^{(n)}) (t)

and

\tilde{W} (z^{(n)}) (t)

approach the function

e^{(λ + μ) t} .

Moreover, at the “limit point”

Z_{1} (β) = λ + μ,

and

Z_{N} (β) = 0

for every

N > 1 .

Consider the case when sequences

\tilde{x}

and

\tilde{y},

defined by (26) and (25) resp., have only finite number of nonzero elements, i.e.,

\tilde{x}, \tilde{y} \in c_{00} .

Then

(\tilde{y} | \tilde{x}) \in c_{00} (Z_{0}) .

So,

[(\tilde{y} | \tilde{x})] \in M_{0} .

Since functions

{\tilde{W}}_{n},

used in the representations (24) and (27) of partition functions are supersymmetric, it follows that values

{\tilde{W}}_{n} (u)

do not depend on the choice of the representative

u \in [(\tilde{y} | \tilde{x})] .

So, it is natural to consider partition functions as functions on such equivalence classes. Note that all elements of the sequence

\tilde{x}

are non-negative and all elements of the sequence

\tilde{y}

are non-positive. So, the equivalence class

[(\tilde{y} | \tilde{x})]

belongs to the subset

M_{0}^{\pm}

M_{0}

defined in the following way. Let us denote by

M_{0}^{\pm}

the set of elements

[u],

where u is of the form

u = (- y_{1}, \dots, - y_{m} | x_{1}, \dots, x_{n}), x_{i} \geq 0, y_{j} \geq 0 .

Note that

M_{0}^{\pm}

can be completed with respect to a ring norm on

M_{0}

(see [15,17]). In SubSection 4.2 we consider such completions more detailed.

For every

[u] \in M_{0}^{\pm}

and odd number

k,

T_{k} (u) = F_{k} (x) + F_{k} (y) \geq 0,

where

x = (x_{1}, \dots, x_{n})

and

y = (y_{1}, \dots, y_{m}),

and it is equal to zero if and only if

u = 0 .

It is known that

M_{0}

contains divisors of zero. For example,

\begin{matrix} [(- 1 | 1)] [(0 | 1, - 1)] & = [((- 1) ⋄ (1, - 1)) • ((1) ⋄ (0)) | ((1) ⋄ (1, - 1)) • ((- 1) ⋄ (0))] \\ = [(- 1, 1) • (0) | (1, - 1) • (0)] \\ = [(- 1, 1, 0 | 1, - 1, 0)] \\ = [(0 | 0)] \\ = 0 . \end{matrix}

Proposition 1.

The set

M_{0}^{\pm}

is a commutative semiring with respect to the ring operations in

M_{0},

without divisors of zero.

Proof.

It is easy to check that if

[u]

and

[v]

are in

M_{0}^{\pm},

then both

[u] + [v]

and

[u] [v]

are in

M_{0}^{\pm} .

But for a given

[u] = [(- y_{1}, \dots, - y_{m} | x_{1}, \dots, x_{n})] \in M_{0}^{\pm},

u \neq 0,

the element

- [u] = [(x_{1}, \dots, x_{n} | - y_{1}, \dots, - y_{m})]

does not belong to

M_{0}^{\pm} .

Thus,

M_{0}^{\pm}

is a semiring but not a ring. If

[u] [v] = 0,

then, by (31),

T_{1} (u) T_{1} (v) = 0 .

So either

T_{1} (u) = 0

T_{1} (v) = 0 .

Thus, either

u = 0

v = 0 .

□

The semiring

M_{0}^{\pm}

has the following important property that

[(- y_{1}, \dots, - y_{m} | x_{1}, \dots, x_{n})] = [(- y_{1}^{'}, \dots, - y_{m^{'}}^{'} | x_{1}^{'}, \dots, x_{n^{'}}^{'})]

if and only if

m = m^{'},

n = n^{'}

and there are permutations

σ

and

τ

such that

(- y_{1}, \dots, - y_{m} | x_{1}, \dots, x_{n}) = (- y_{τ (1)}^{'}, \dots, - y_{τ (m)}^{'} | x_{σ (1)}^{'}, \dots, x_{σ (n)}^{'}) .

Let

♯ [u]

be a pair

(m, n)

such that in the representation

[u] = [(- y_{1}, \dots, - y_{m} | x_{1}, \dots, x_{n})]

M_{0}^{\pm}

the number of nonzero elements

y_{j}

is equal to m and the number of nonzero elements

x_{i}

is equal to

n .

From the definition of the ring operations in

M_{0},

we have that if

♯ [u] = (m, n)

and

♯ [v] = (k, s),

then

♯ ([u] + [v]) = (m + k, n + s)

and

♯ ([u] [v]) = (m s + n k, m k + n s) .

In particular,

♯ {[u]}^{2} = (2 m n, m^{2} + n^{2}) .

Proposition 2.

Every invertible element in

M_{0}^{\pm}

is of the form

(0 | x)

for some

x > 0

(- y | 0)

for some

y > 0 .

Every idempotent

[u]

M_{0}^{\pm}

is of the form

[u] = [(0 | 1)]

[u] = [(- 1 | 0)] .

Proof.

Let

[u] [v] = [(0 | 1)],

then

♯ ([u] [v]) = (0, 1)

and, so,

♯ [u] = (0, 1)

and

♯ [v] = (0, 1)

♯ [u] = (1, 0)

and

♯ [v] = (1, 0) .

Consequently,

u = (0 | x)

and

v = (0 | 1 / x)

for some

x > 0

u = (- y | 0)

and

v = (- 1 / y | 0)

for some

y > 0 .

Let

[u]

be an idempotent in

M_{0}^{\pm},

that is,

{[u]}^{r} = [u]

for some positive integer

r > 1 .

Then

♯ {[u]}^{r} = ♯ [u]

only if

♯ [u] = (1, 0)

♯ [u] = (0, 1) .

Elements of the form

[(- a | 0)

and

[(0 | a)],

a > 0,

are idempotents only if

a = 1 .

□

Proposition 3.

Elements of the form

[(- x_{1}, \dots, - x_{n} | x_{1}, \dots, x_{n})],

x_{i} > 0

can be represented as

[(- x_{1}, \dots, - x_{n} | x_{1}, \dots, x_{n})] = [(- 1 | 1)] [(- x_{1}, \dots, - x_{k} | x_{k + 1}, \dots, x_{n})]

for every integer

k,

0 \leq k \leq n .

Proof.

The straightforward computation. □

From the proposition it follows that we have no multiplicative cancelation in

M_{0}^{\pm},

that is, the equalities

[u] [v] = [w]

and

[u] [v^{'}] = [w]

do not imply

[v] = [v^{'}] .

4.2. A tropical semiring structure

We introduce another semiring structure on

M_{0}^{\pm}

which is related to Tropical Mathematics. Some applications of tropical semirings to Quantum Mechanics can be found in [34]. Let us recall that the min tropical semiring is the semiring

(R \cup {+ \infty}, \underset{̲}{\oplus}, ⊙),

where the operations

\underset{̲}{\oplus}

and ⊙ are defined by

x \underset{̲}{\oplus} y = min {x, y}, x ⊙ y = x + y, x, y \in R \cup {+ \infty} .

The operations

\underset{̲}{\oplus}

and ⊙ are called the tropical addition and the tropical multiplication respectively. The unit for

\underset{̲}{\oplus}

+ \infty,

and the unit for ⊙ is

0 .

Similarly, the max tropical semiring is the semiring

(R \cup {- \infty}, \bar{\oplus}, ⊙)

such that

x \bar{\oplus} y = max {x, y}, x ⊙ y = x + y, x, y \in R \cup {- \infty} .

In this semiring, the unit for

\bar{\oplus}

- \infty

and the unit for ⊙ is

0 .

The semirings are isomorphic with respect to the mapping

x \mapsto - x .

The usual metric

ρ (a, b) = | a - b |

R

can be extended to

R \cup {- \infty}

by setting

ρ (a, - \infty) = 1

for every

a \in R .

Similarly,

ρ (a, + \infty) = 1,

a \in R,

for the case

R \cup {+ \infty} .

Let

(- y_{m}, \dots, - y_{1} | x_{1}, \dots, x_{n})

be a representation of

[u] \in M_{0}^{\pm} .

We say that this representation is ordered if

x_{1} \geq x_{2} \geq \dots \geq x_{n}

and

- y_{1} \leq - y_{2} \leq \dots \leq - y_{m} .

The ordered representation of

[u]

is unique and we denote it by

{(- y_{m}, \dots, - y_{1} | x_{1}, \dots, x_{n})}_{o} .

Let us denote by

e

the formal element

e = (\dots, + \infty, \dots, + \infty, + \infty | - \infty, - \infty, \dots, - \infty, \dots) .

Definition 2.

Let us define a tropical semiring

M_{0}^{\oplus}

as the set

M_{0}^{\pm} \cup {e}

with operations ⊕ and ⊙ such that

\begin{matrix} (- y_{m}, & \dots, - y_{1} {| x_{1}, \dots, x_{n})}_{o} \oplus {(- d_{m}, \dots, - d_{1} | b_{1}, \dots, b_{n})}_{o} \\ = {((- y_{m}) \underset{̲}{\oplus} (- d_{m}), \dots, (- y_{1}) \underset{̲}{\oplus} (- d_{1}) | x_{1} \bar{\oplus} b_{1}, \dots, x_{n} \bar{\oplus} b_{n})}_{o} \\ = {(min {- y_{m}, - d_{m}}, \dots, min {- y_{1}, - d_{1}} | max {x_{1}, b_{1}}, \dots, max {x_{n}, b_{n}})}_{o} \end{matrix}

and

\begin{matrix} (- y_{m}, & \dots, - y_{1} {| x_{1}, \dots, x_{n})}_{o} ⊙ {(- d_{m}, \dots, - d_{1} | b_{1}, \dots, b_{n})}_{o} \\ = {((- y_{m}) ⊙ (- d_{m}), \dots, (- y_{1}) ⊙ (- d_{1}) | x_{1} ⊙ b_{1}, \dots, x_{n} ⊙ b_{n})}_{o} \\ = {(- y_{m} - d_{m}, \dots, - y_{1} - d_{1} | x_{1} + b_{1}, \dots, x_{n} + b_{n})}_{o} . \end{matrix}

Proposition 4.

M_{0}^{\pm} \cup {e}

is a semiring and the unit for ⊕ is

e

and the unit for ⊙ is

0 .

Proof.

Let us check the distributive law. From the distributive laws in the min tropical semiring and in the max tropical semiring,

{(- c_{m}, \dots, - c_{1} | a_{1}, \dots, a_{n})}_{o} ⊙ ({(- y_{m}, \dots, - y_{1} | x_{1}, \dots, x_{n})}_{o} \oplus {(- d_{m}, \dots, - d_{1} | b_{1}, \dots, b_{n})}_{o})

= {(- c_{m} ⊙ ((- y_{m}) \underset{̲}{\oplus} (- d_{m})), \dots, - c_{1} ⊙ ((- y_{1}) \underset{̲}{\oplus} (- d_{1})) | a_{1} ⊙ (x_{1} \bar{\oplus} b_{1}), \dots, a_{n} ⊙ (x_{n} \bar{\oplus} b_{n}))}_{o}

= {(- c_{m}, \dots, - c_{1} | a_{1}, \dots, a_{n})}_{o} ⊙ {(- y_{m}, \dots, - y_{1} | x_{1}, \dots, x_{n})}_{o}

\oplus {(- c_{m}, \dots, - c_{1} | a_{1}, \dots, a_{n})}_{o} ⊙ {(- d_{m}, \dots, - d_{1} | b_{1}, \dots, b_{n})}_{o} .

□

Let X be a Banach space with an unconditional Schauder basis

(e_{n}),

n \in N .

Then any vector

x \in X

can be represented as

x = \sum_{n = 1}^{\infty} x_{n} e_{n} = (x_{1}, \dots, x_{n}, \dots) .

Denote by

M_{X}

the ring of elements

[u] = [(\dots, y_{m}, \dots, y_{1} | x_{1}, \dots, x_{n}, \dots)], x_{i}, y_{j}, \in C

such that

x = (x_{1}, \dots, x_{n}, \dots)

and

y = (y_{1}, \dots, y_{m}, \dots)

are in X endowed with the following ring norm

∥ [u] ∥ = inf ({∥ x ∥}_{X} + {∥ y ∥}_{X}),

where the infimum is taken over all representations

u = (\dots, y_{m}, \dots, y_{1} | x_{1}, \dots, x_{n}, \dots) .

It is known that this norm generates a metric

d ([u], [v]) = ∥ [u] - [v] ∥

and

M_{X}

is a complete metric space with respect to the metric. Moreover, the ring operations in

M_{X}

are continuous and

M_{0}

is a dense subring in

M_{X}

[15,17].

Let us denote by

M_{X}^{\pm}

the closed subset in

M_{X}

consisting of elements

[u] = [(\dots, - y_{m}, \dots, - y_{1} | x_{1}, \dots, x_{n}, \dots)], x_{i} \geq 0, y_{j} \geq 0 .

Thus

M_{X}^{\pm}

is a complete metric space and a topological semiring.

We can extend the metric to

M_{X}^{\oplus} = M_{X}^{\pm} \cup {e}

by setting

d (u, e) = 1

for every

u \in M_{X}^{\oplus} .

Note that

M_{X}^{\oplus} ∖ {e}

is a commutative group with respect to “⊙” and

∥ {[u]}^{⊙ k} ∥ = ∥ \underset{k}{\underset{︸}{[u] ⊙ \dots ⊙ [u]}} ∥ = ∥ [k u] ∥ = k ∥ u ∥, k \in N .

Theorem 1.

For any Banach space X with an unconditional basis the following statements are true:

1: The tropical operations are continuous in $M_{X}^{\oplus}$ ;
2: The mappings

$Φ^{+} [u] = max_{i} x_{i} a n d Φ^{-} [u] = min_{j} (- y_{j})$

are continuous semiring homomorphisms from $M_{X}^{\oplus}$ to the max tropical semiring $(R \cup {+ \infty}, \bar{\oplus}, ⊙)$ and to the min tropical semiring $(R \cup {- \infty}, \underset{̲}{\oplus}, ⊙)$ respectively.

Proof.

1. If

[u]

and

[v]

are not equal to

e,

then

∥ [u] \oplus [v] ∥ \leq ∥ [u] ⊙ [v] ∥ \leq ∥ [u] • [v] ∥

and we know that the operation “•” is continuous.

2. Clearly,

Φ^{+} ([u]) = x_{1}

and

Φ^{-} ([u]) = - y_{1},

in particular,

Φ^{+} (e) = + \infty,

and

Φ^{-} (e) = - \infty .

Also,

Φ^{+} ([u] \oplus [u^{'}]) = max (x_{1}, x_{1}^{'}) = Φ^{+} ([u]) \oplus Φ^{+} ([u^{'}])

and

Φ^{+} ([u] ⊙ [u^{'}]) = x_{1} + x_{1}^{'} = Φ^{+} ([u]) ⊙ Φ^{+} ([u^{'}]) .

Thus

Φ^{+}

is a semiring isomorphism.

To show the continuity, we observe that the function

x = (x_{1}, x_{2}, \dots) \mapsto {max}_{n} | x_{n} |

is bounded (on bounded subsets) on every Banach space X with a Schauder basis

(e_{n}) .

Indeed, if

(π_{n}),

n \in N

be the sequence of projections,

π_{n} (x) = \sum_{k = 1}^{n} x_{k} e_{k},

then

sup_{n} ∥ π_{n} ∥ = K < \infty

(see [35], pp. 1-2) and so

| x_{n} | = | π_{n + 1} (x) - π_{n} {(x) | \leq 2 K ∥ x ∥}_{X} .

Hence

| Φ^{+} ([u]) | \leq 2 K ∥ [u] ∥ .

The continuity of

Φ^{+}

follows from ([36], Theorem 11.22) taking into account that

Φ^{+}

is a bounded homomorphism of the multiplicative normed group

M_{X}^{\oplus} ∖ e

such that

∥ {[u]}^{⊙ k} ∥ = k ∥ u ∥,

k \in N .

The same works for

Φ^{-} .

□

5. Discussions and Conclusions

In this paper we continue to develop the ideas proposed by Schmidt and Schnack in [2,3] about involving symmetric polynomials for investigations of the partition functions of ideal quantum gases. The first goal of the paper was to find a correspondence between algebraic bases of supersymmetric polynomials and partition functions of ideal gases consisting of both bosons and fermions. We can see that combinatorial relations in the algebra of supersymmetric polynomials have corresponding physical interpretations. Taking into account that two elements (vectors) z and

z^{'},

in the set of possible energy levels are equivalent if and only if

P (z) = P (z^{'})

for every supersymmetric polynomial

P,

it is naturally to consider the quotient set with respect to the equivalence as a natural domain. For such a quotient set the usual vector operations are not valid and we introduced new ring operations (addition and multiplication) on the quotient set

M_{0} .

It seems to be that the new addition can be obtained using the direct sum of operators

exp (- β H_{1})

and

exp (- β H_{2}),

while the new product leads to the tensor product of operators. Note that the elements of

M_{0}

have the physical interpretation if

x_{i} \geq 0

and

y_{j} \leq 0 .

Otherwise, we can get a system where the cancelation rule

[(y, a | a, x)] = [(x | y)]

plays a non-trivial role, and where we can get a negative energy. It leads us to tachyonic particles that cannot exist because they are inconsistent with the known laws of physics. But such an approach can be interesting for tachyon condensation (for details on tachyon condensation see [37]).

The fact that the energy on a level can not be negative suggests using elements in

M_{0}

which have very specific form,

[z] = [(- y_{1},, - y_{m} | x_{1}, \dots, x_{n})],

where all

x_{i}

and

y_{j}

are non-negative. The subset of such elements forms a semiring without divisions of zero, denoted by

{M_{0}}^{\pm} .

We considered algebraic properties of this semiring and its completions

M_{X}^{\pm}

with respect to various metrics associated with different Banach spaces

X .

Also, we introduced new operations on

M_{X}^{\oplus} = M_{X}^{\pm} \cup e

that lead to an infinite-dimensional analog of the so-called tropical semirings. We proved the continuity of the operations on

M_{X}^{\oplus}

and constructed some real-valued homomorphisms of

M_{X}^{\oplus} .

For further investigation we are going to use block-symmetric (or MacMahon) and block-supersymmetric polynomials on

ℓ_{1} (C^{s})

and their applications to partition functions of quantum gases. The space

ℓ_{1} (C^{s})

can be defined as a vector space of sequences

x = (x^{(1)}, \dots, x^{(n)}, \dots)

such that every element

x^{(n)} = (x_{1}^{(n)}, \dots, x_{s}^{(n)})

is a vector in

C^{s},

and

∥ x ∥ = \sum_{n = 1}^{\infty} ∥ x^{(n)} ∥ .

A polynomial is block-symmetric on

ℓ_{1} (C^{s})

if it is symmetric with respect to all permutations of the vectors (blocks)

x^{(n)} .

We can expect that models based on block-symmetric (and maybe block-supersymmetric) polynomials can be useful for describing quantum gases with entanglement particles.

Combinatorial properties of block-symmetric polynomials were considered in [38]. Algebras of block-symmetric polynomials and analytic functions and corresponding bases of polynomials on

ℓ_{1} (C^{s})

were studied in [39,40,41,42,43,44]. Applications of block-symmetric polynomials for the quantum product of symmetric functions were proposed in [45].

Author Contributions

Conceptualization, A.Z. and T.V.; investigation, I.C. and M.M.; writing—original draft preparation, I.C. and T.V.; writing—review and editing, A.Z.; project administration, A.Z. All authors have read and agreed to the published version of the manuscript.

Funding

This research was supported by the National Research Foundation of Ukraine, 2020.02/0025.

Data Availability Statement

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

Conflicts of Interest

The authors declare no conflict of interest.

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Applications of Supersymmetric Polynomials in Statistical Quantum Physics

Abstract

1. Introduction

2. Preliminaries results on symmetric polynomials and partition functions

2.1. Symmetric polynomials

2.2. Partition functions

2.3. Note about the Banach space ℓ 1

3. Supersymmetric polynomials and partition functions for mixed systems of bosons and fermions

4. Semiring structures on the set of variables

4.1. The ring M 0

4.2. A tropical semiring structure

5. Discussions and Conclusions

Author Contributions

Funding

Data Availability Statement

Conflicts of Interest

References

MDPI Initiatives

Important Links

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2.3. Note about the Banach space $ℓ_{1}$

4.1. The ring $M_{0}$