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(−1)F
In a quantum field theory with fermions, (−1)''F'' is a unitary operator, unitary, Hermitian operator, Hermitian, Involution (mathematics), involutive Operator (mathematics), operator where ''F'' is the fermion number operator. For the example of particles in the Standard Model, it is equal to the sum of the lepton number plus the baryon number, . The action of this operator is to multiply bosonic states by 1 and fermionic states by −1. This is always a global internal symmetry of any quantum field theory with fermions and corresponds to a rotation by 2π. This splits the Hilbert space into two superselection sectors. Bosonic operators Commutativity, commute with (−1)''F'' whereas fermionic operators anticommute with it. This operator really shows its utility in supersymmetry, supersymmetric theories. Witten index, Its trace is the spectral asymmetry of the fermion spectrum, and can be understood physically as the Casimir effect. See also *Parity (physics) *Primon gas *Mà ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (physics)
In physics, a parity transformation (also called parity inversion) is the flip in the sign of ''one'' spatial coordinate. In three dimensions, it can also refer to the simultaneous flip in the sign of all three spatial coordinates (a point reflection): :\mathbf: \beginx\\y\\z\end \mapsto \begin-x\\-y\\-z\end. It can also be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity. The weak interaction is chiral and thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. A matrix representation of P (in any number of dimensions) has determinant equal to −1, and hence is distinct from a rotat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casimir Effect
In quantum field theory, the Casimir effect is a physical force acting on the macroscopic boundaries of a confined space which arises from the quantum fluctuations of the field. It is named after the Dutch physicist Hendrik Casimir, who predicted the effect for electromagnetic systems in 1948. In the same year, Casimir together with Dirk Polder described a similar effect experienced by a neutral atom in the vicinity of a macroscopic interface which is referred to as the Casimir–Polder force. Their result is a generalization of the London–van der Waals force and includes retardation due to the finite speed of light. Since the fundamental principles leading to the London–van der Waals force, the Casimir and the Casimir–Polder force, respectively, can be formulated on the same footing, the distinction in nomenclature nowadays serves a historical purpose mostly and usually refers to the different physical setups. It was not until 1997 that a direct experiment by S. La ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Operator
In quantum mechanics, for systems where the total number of particles may not be preserved, the number operator is the observable that counts the number of particles. The number operator acts on Fock space. Let :, \Psi\rangle_\nu=, \phi_1,\phi_2,\cdots,\phi_n\rangle_\nu be a Fock state, composed of single-particle states , \phi_i\rangle drawn from a basis of the underlying Hilbert space of the Fock space. Given the corresponding creation and annihilation operators a^(\phi_i) and a(\phi_i)\, we define the number operator by :\hat \ \stackrel\ a^(\phi_i)a(\phi_i) and we have :\hat, \Psi\rangle_\nu=N_i, \Psi\rangle_\nu where N_i is the number of particles in state , \phi_i\rangle. The above equality can be proven by noting that :\begin a(\phi_i) , \phi_1,\phi_2,\cdots,\phi_,\phi_i,\phi_,\cdots,\phi_n\rangle_\nu &=& \sqrt , \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_n\rangle_\nu \\ a^(\phi_i) , \phi_1,\phi_2,\cdots,\phi_,\phi_,\cdots,\phi_n\rangle_\nu &=& \sqrt , \phi_1,\phi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Field Theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. QFT treats particles as excited states (also called Quantum, quanta) of their underlying quantum field (physics), fields, which are more fundamental than the particles. The equation of motion of the particle is determined by minimization of the Lagrangian, a functional of fields associated with the particle. Interactions between particles are described by interaction terms in the Lagrangian (field theory), Lagrangian involving their corresponding quantum fields. Each interaction can be visually represented by Feynman diagrams according to perturbation theory (quantum mechanics), perturbation theory in quantum mechanics. History Quantum field theory emerged from the wo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anticommute
In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the ''inverse'' of the result with unswapped arguments. The notion '' inverse'' refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of gives for example, Another prominent example of an anticommutative operation is the Lie bracket. In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments. Definition If A, B are two abelian groups, a bilinear map f\colon A^2 \to B is anticommutative if for all x, y \in A we have :f(x, y) = - f(y, x). More generally, a multilinear map g : A^n \to B is anticommutative if for all x_1, \d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Function
The Möbius function is a multiplicative function in number theory introduced by the German mathematician August Ferdinand Möbius (also transliterated ''Moebius'') in 1832. It is ubiquitous in elementary and analytic number theory and most often appears as part of its namesake the Möbius inversion formula. Following work of Gian-Carlo Rota in the 1960s, generalizations of the Möbius function were introduced into combinatorics, and are similarly denoted . Definition For any positive integer , define as the sum of the primitive th roots of unity. It has values in depending on the factorization of into prime factors: * if is a square-free positive integer with an even number of prime factors. * if is a square-free positive integer with an odd number of prime factors. * if has a squared prime factor. The Möbius function can alternatively be represented as : \mu(n) = \delta_ \lambda(n), where is the Kronecker delta, is the Liouville function, is the number of dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primon Gas
In mathematical physics, the primon gas or free Riemann gas is a toy model illustrating in a simple way some correspondences between number theory and ideas in quantum field theory and dynamical systems. It is a quantum field theory of a set of non-interacting particles, the primons; it is called a gas or a ''free model'' because the particles are non-interacting. The idea of the primon gas was independently discovered by Donald Spector and Bernard Julia. Later works by Bakas and Bowick and Spector D. Spector, Duality, Partial Supersymmetry, and Arithmetic Number Theory, J. Math. Phys. 39 (1998) pp. 1919–1927 explored the connection of such systems to string theory. The model State space Consider a Hilbert space H with an orthonormal basis of states , p\rangle labelled by the prime numbers ''p''. Second quantization gives a new Hilbert space K, the bosonic Fock space on H, where states describe collections of primes - which we can call primons if we think of them as analogous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Asymmetry
In mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator. In mathematics, the spectral asymmetry arises in the study of elliptic operators on compact manifolds, and is given a deep meaning by the Atiyah-Singer index theorem. In physics, it has numerous applications, typically resulting in a fractional charge due to the asymmetry of the spectrum of a Dirac operator. For example, the vacuum expectation value of the baryon number is given by the spectral asymmetry of the Hamiltonian operator. The spectral asymmetry of the confined quark fields is an important property of the chiral bag model. For fermions, it is known as the Witten index, and can be understood as describing the Casimir effect for fermions. Definition Given an operator with eigenvalues \omega_n, an equal number of which are positive and negative, the spectral asymmetry may be defined as the sum :B=\lim_ \frac\sum_n \sgn(\omega_n) \exp (- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witten Index
In quantum field theory and statistical mechanics, the Witten index at the inverse temperature β is defined as a modification of the standard partition function: :\textrm -1)^F e^/math> Note the (-1)F operator, where F is the fermion number operator. This is what makes it different from the ordinary partition function. It is sometimes referred to as the spectral asymmetry. In a supersymmetric theory, each nonzero energy eigenvalue contains an equal number of bosonic and fermionic states. Because of this, the Witten index is independent of the temperature and gives the number of zero energy bosonic vacuum states minus the number of zero energy fermionic vacuum states. In particular, if supersymmetry is spontaneously broken then there are no zero energy ground states and so the Witten index is equal to zero. The Witten index of the supersymmetric sigma model on a manifold is given by the manifold's Euler characteristic.* p191 (10.124) :\textrm -1)^F e^\sum_(-1)^pb_p=\ch ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Supersymmetry
In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories exist. Supersymmetry is a spacetime symmetry between two basic classes of particles: bosons, which have an integer-valued spin and follow Bose–Einstein statistics, and fermions, which have a half-integer-valued spin and follow Fermi–Dirac statistics. In supersymmetry, each particle from one class would have an associated particle in the other, known as its superpartner, the spin of which differs by a half-integer. For example, if the electron exists in a supersymmetric theory, then there would be a particle called a ''"selectron"'' (superpartner electron), a bosonic partner of the electron. In the simplest supersymmetry theories, with perfectly " unbroken" supersymmetry, each pair of superpartners would share the same mass and intern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press
Oxford University Press (OUP) is the university press of the University of Oxford. It is the largest university press in the world, and its printing history dates back to the 1480s. Having been officially granted the legal right to print books by decree in 1586, it is the second oldest university press after Cambridge University Press. It is a department of the University of Oxford and is governed by a group of 15 academics known as the Delegates of the Press, who are appointed by the vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, opposite Somerville College, in the inner suburb of Jericho. For the last 500 years, OUP has primarily focused on the publication of pedagogical texts and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 originally introduced by Wick, Wightman, and Wigner to impose additional restrictions to quantum theory beyond those of selection rules. Mathematically speaking, two quantum states \psi_1 and \psi_2 are separated by a selection rule if \langle \psi_1 , H , \psi_2 \rangle = 0 for the given Hamiltonian H , while they are separated by a superselection rule if \langle \psi_1 , A , \psi_2 \rangle = 0 for ''all ''physical observables A . Because no observable connects \langle \psi_1 , and , \psi_2 \rangle they cannot be put into a quantum superposition \alpha , \psi_1 \rangle + \beta , \psi_2 \rangle , and/or a quantum superposition cannot be distinguished from a classical mixture of the two states. It also implies that there is a class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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