Furry's Theorem
   HOME

TheInfoList



OR:

In
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
, Furry's theorem states that if a
Feynman diagram In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced ...
consists of a closed loop of
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 ...
lines with an odd number of vertices, its contribution to the
amplitude The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
vanishes. As a
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving another ...
, a single
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
cannot arise from the
vacuum A vacuum (: vacuums or vacua) is space devoid of matter. The word is derived from the Latin adjective (neuter ) meaning "vacant" or "void". An approximation to such vacuum is a region with a gaseous pressure much less than atmospheric pressur ...
or be absorbed by it. The theorem was first derived by Wendell H. Furry in 1937, as a direct consequence of the
conservation of energy The law of conservation of energy states that the total energy of an isolated system remains constant; it is said to be Conservation law, ''conserved'' over time. In the case of a Closed system#In thermodynamics, closed system, the principle s ...
and charge conjugation symmetry.


Theory

Quantum electrodynamics has a number of
symmetries Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
, one of them being the
discrete symmetry In mathematics and geometry, a discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square ...
of charge conjugation. This acts on fields through 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 ...
charge conjugation operator C which anticommutes with the photon field A_\mu(x) as CA^\mu(x) C^\dagger = -A^\mu(x), while leaving the vacuum state invariant C, \Omega\rangle = , \Omega\rangle. Considering the simplest case of the
correlation function A correlation function is a function that gives the statistical correlation between random variables, contingent on the spatial or temporal distance between those variables. If one considers the correlation function between random variables ...
of a single photon operator gives : \langle \Omega, A^\mu(x), \Omega\rangle = \langle \Omega, C^\dagger C A^\mu(x) C^\dagger C, \Omega\rangle = - \langle \Omega, A^\mu(x), \Omega\rangle, so this correlation function must vanish. For n photon operators, this argument shows that under charge conjugation this picks up a factor of (-1)^n and thus vanishes when n is odd. More generally, since the charge conjugation operator also anticommutes with the vector current j^\mu(x), Furry's theorem states that the correlation function of any odd number of on-shell or off-shell photon fields and/or currents must vanish in quantum electrodynamics. Since the theorem holds at the
non-perturbative In mathematics and physics, a non-perturbative function (mathematics), function or process is one that cannot be described by perturbation theory. An example is the function : f(x) = e^, which does not equal its own Taylor series in any neighbo ...
level, it must also hold at each order in
perturbation theory In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
. At leading order this means that any fermion loop with an odd number of vertices must have a vanishing contribution to the amplitude. An explicit calculation of these diagrams reveals that this is because the diagram with a fermion going clockwise around the loop cancels with the second diagram where the fermion goes anticlockwise. The vanishing of the three vertex loop can also be seen as a consequence of the
renormalizability Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
of quantum electrodynamics since the bare
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
does not have any counterterms involving three photons.


Applications and limitations

Furry's theorem allows for the simplification of a number of amplitude calculations in quantum electrodynamics. In particular, since the result also holds when photons are off-shell, all Feynman diagrams which have at least one internal fermion loops with an odd number of vertices have a vanishing contribution to the amplitude and can be ignored. Historically the theorem was important in showing that the scattering of photons by an external field, known as Delbrück scattering, does not proceed via a triangle diagram and must instead proceed through a box diagram. In the presence of a background charge density or a nonzero
chemical potential In thermodynamics, the chemical potential of a Chemical specie, species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potent ...
, Furry's theorem is broken, although if both these vanish then it does hold at nonzero temperatures as well as at zero temperatures. It also does not apply in the presence of a strong background
magnetic field A magnetic field (sometimes called B-field) is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular ...
where photon splitting interactions \gamma \rightarrow \gamma \gamma are allowed, a process that may be detected in
astrophysical Astrophysics is a science that employs the methods and principles of physics and chemistry in the study of astronomical objects and phenomena. As one of the founders of the discipline, James Keeler, said, astrophysics "seeks to ascertain the ...
settings such as around
neutron star A neutron star is the gravitationally collapsed Stellar core, core of a massive supergiant star. It results from the supernova explosion of a stellar evolution#Massive star, massive star—combined with gravitational collapse—that compresses ...
s. The theorem also does not hold when Weyl fermions are involved in the loops rather than
Dirac fermion In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. A vast majority of fermions fall under this category. Description In particle physics, all fermions in the standard model have distinct antipar ...
s, resulting in non-vanishing odd vertex number diagrams. In particular, the non-vanishing of the triangle diagram with Weyl fermions gives rise to the
chiral anomaly In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is analogous to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have mor ...
, with the sum of these having to cancel for a quantum theory to be consistent. While the theorem has been formulated in quantum electrodynamics, a version of it holds more generally. For example, while 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 ...
is not charge conjugation invariant due to
weak interaction In nuclear physics and particle physics, the weak interaction, weak force or the weak nuclear force, is one of the four known fundamental interactions, with the others being electromagnetism, the strong interaction, and gravitation. It is th ...
s, the fermion loop diagrams with an odd number of photons attached will still vanish since these are equivalent to a purely quantum electrodynamical diagram. Similarly, any diagram involving such loops as sub-diagrams will also vanish. It is however no longer true that all odd number photon diagrams need to vanish. For example, relaxing the requirement of charge conjugation and parity invariance of quantum electrodynamics, as occurs when weak interactions are included, allows for a three-photon vertex term. While this term does give rise to \gamma \rightarrow \gamma \gamma interactions, they only occur if two of the photons are virtual; searching for such interactions must be done indirectly, such as through
bremsstrahlung In particle physics, bremsstrahlung (; ; ) is electromagnetic radiation produced by the deceleration of a charged particle when deflected by another charged particle, typically an electron by an atomic nucleus. The moving particle loses kinetic ...
experiments from electron-positron collisions. In non-Abelian Yang–Mills theories, Furry's theorem does not hold since these involve noncommuting
color charge Color charge is a property of quarks and gluons that is related to the particles' strong interactions in the theory of quantum chromodynamics (QCD). Like electric charge, it determines how quarks and gluons interact through the strong force; ho ...
s. For example, the quark triangle diagrams with three external
gluon A gluon ( ) is a type of Massless particle, massless elementary particle that mediates the strong interaction between quarks, acting as the exchange particle for the interaction. Gluons are massless vector bosons, thereby having a Spin (physi ...
s are proportional to two different generator
traces Traces may refer to: Literature * ''Traces'' (book), a 1998 short-story collection by Stephen Baxter * ''Traces'' series, a series of novels by Malcolm Rose Music Albums * ''Traces'' (Classics IV album) or the title song (see below), 1969 * ''Tra ...
\text ^aT^bT^c\neq \text ^aT^cT^b/math> and so they do not cancel. However, charge conjugation arguments can still be applied in limited cases such as to deduce that the triangle diagram gg \rightarrow X for a color neutral
spin Spin or spinning most often refers to: * Spin (physics) or particle spin, a fundamental property of elementary particles * Spin quantum number, a number which defines the value of a particle's spin * Spinning (textiles), the creation of yarn or thr ...
1^-
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 ...
vanishes.


See also

*
Landau–Yang theorem In quantum mechanics, the Landau–Yang theorem is a selection rule for particles that decay into two on-shell photons. The theorem states that a massive particle with spin 1 cannot decay into two photons. Assumptions A photon here is any part ...
*
Ward–Takahashi identity In quantum field theory, a Ward–Takahashi identity is an identity between correlation functions that follows from the global or gauge symmetries of the theory, and which remains valid after renormalization. The Ward–Takahashi identity of qua ...
*
Wick's theorem Wick's theorem is a method of reducing high- order derivatives to a combinatorics problem. It is named after Italian physicist Gian Carlo Wick. It is used extensively in quantum field theory to reduce arbitrary products of creation and annihil ...


References

{{QED Quantum electrodynamics Scattering theory Theorems in quantum mechanics