Two particles are said to be identical if all their intrinsic properties, e.g., mass, electrical charge, spin, colour are exactly the same. Similarly, all protons are identical to one another, and so are all quarks. They possess the same defining properties under similar interactions with each other. The exchange in position of identical particles has a so-called permutation symmetry. Clearly, this permutation symmetry is extremely important for quantum mechanical calculations of the multi-indistinguishable particles. So knowing how a collection of fermions (or bosons) behave is kinematically determined by their permutation symmetry. If we have different cycles in a permutation and there is no element common to any two different cycles, then all these cycles are termed disjoint. In fact any permutation of a finite set is the product of disjoint cycles. A cycle of length two is called a transposition.Two particles are said to be identical if all their intrinsic properties, e.g., mass, electrical charge, spin, colour are exactly the same. Similarly, all protons are identical to one another, and so are all quarks. They possess the same defining properties under similar interactions with each other. The exchange in position of identical particles has a so-called permutation symmetry. Clearly, this permutation symmetry is extremely important for quantum mechanical calculations of the multi-indistinguishable particles. So knowing how a collection of fermions (or bosons) behave is kinematically determined by their permutation symmetry. If we have different cycles in a permutation and there is no element common to any two different cycles, then all these cycles are termed disjoint. In fact any permutation of a finite set is the product of disjoint cycles. A cycle of length two is called a transposition.

CRC Press eBooksAug 17, 20160 citations

Generalized dirac identities and explicit relations between the permutational symmetry and the spin operators for systems of identical particles

Jacob Katriel, Josef Paldus, Reuben Pauncz

International Journal of Quantum ChemistryJan 1, 19861 citation

The dynamical instability of nonrelativistic many-body systems

Charles Radin

Communications in Mathematical PhysicsFeb 1, 197715 citations

Permutational symmetry for identical multilevel systems: A second-quantized approach

Rui E. F. Silva, Johannes Feist

We develop a framework that provides a straightforward approach to fully exploit the permutational symmetry of identical multilevel systems. By taking into account the permutational symmetry, we outline a simple scheme that allows us to map the dynamics of $N$ identical $d$ -level systems to the dynamics of $d$ bosonic modes with $N$ particles, achieving an exponential reduction on the dimensionality of the problem in a simple and straightforward way. In particular, we consider the Lindblad dynamics of several identical multilevel systems interacting with a common subsystem under the action of collective dissipation terms.We develop a framework that provides a straightforward approach to fully exploit the permutational symmetry of identical multilevel systems. By taking into account the permutational symmetry, we outline a simple scheme that allows us to map the dynamics of $N$ identical $d$-level systems to the dynamics of $d$ bosonic modes with $N$ particles, achieving an exponential reduction on the dimensionality of the problem in a simple and straightforward way. In particular, we consider the Lindblad dynamics of several identical multilevel systems interacting with a common subsystem under the action of collective dissipation terms.

arXiv:2201.07611Physical Review AApr 4, 202220 citations

N

identical

d

-level systems to the dynamics of

d

bosonic modes with

N

particles, achieving an exponential reduction on the dimensionality of the problem in a simple and straightforward way. In particular, we consider the Lindblad dynamics of several identical multilevel systems interacting with a common subsystem under the action of collective dissipation terms.