In a

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

with

fermion In particle physics, a fermion is a subatomic particle that follows Fermi–Dirac statistics. Fermions have a half-integer spin (spin 1/2, spin , Spin (physics)#Higher spins, spin , etc.) and obey the Pauli exclusion principle. These particles i ...

s, (−1)^''F'' is a

unitary Unitary may refer to: Mathematics * Unitary divisor * Unitary element * Unitary group * Unitary matrix * Unitary morphism * Unitary operator * Unitary transformation * Unitary representation * Unitarity (physics) * ''E''-unitary inverse semigr ...

Hermitian {{Short description, none Numerous things are named after the French mathematician Charles Hermite (1822–1901): Hermite * Cubic Hermite spline, a type of third-degree spline * Gauss–Hermite quadrature, an extension of Gaussian quadrature me ...

, involutive

operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...

where ''F'' is the

number operator. For the example of particles in the

Standard Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...

, it is equal to the sum of the lepton number plus the baryon number, . The action of this operator is to multiply

boson In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-intege ...

ic states by 1 and

ic states by −1. This is always a global

internal symmetry The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuous'' (such ...

of any quantum field theory with fermions and corresponds to a rotation by 2π. This splits the

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

into two

superselection sector In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence between eigenstates of certain observables. It was origina ...

s. Bosonic operators commute with (−1)^''F'' whereas fermionic operators anticommute with it. This operator really shows its utility in

supersymmetric Supersymmetry is a theoretical framework in physics that suggests the existence of a symmetry between particles with integer spin (''bosons'') and particles with half-integer spin (''fermions''). It proposes that for every known particle, there ...

theories. Its trace is the spectral asymmetry of the fermion spectrum, and can be understood physically as the

Casimir effect In quantum field theory, the Casimir effect (or Casimir force) is a physical force (physics), force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of a field (physics), field. The term Casim ...

See also

References

Further reading