Massless field

From Encyclopedia of Mathematics
Jump to: navigation, search

A quantum field theory is said to contain massless fields if in the Hilbert space generated by repeated application of the quantum fields to the vacuum state there exist subspaces associated with one-particle states of mass zero.

According to the concept of a relativistic particle, introduced by E. Wigner [a1], the one-particle states associated with a particle of type $[ m , s ]$ are given by Hilbert subspaces transforming irreducibly under the unitary representation of the covering group of the orthochronous proper Poincaré group (cf. also Poincaré group) $\widetilde { \mathcal{P} }_+ ^ { \uparrow }$. Here, $m$ is the eigenvalue of the states in these subspaces with respect to the mass operator $M = \sqrt { P _ { \mu } P ^ { \mu } }$, where the $P ^ { \mu }$ ($\mu = 0,1,2,3$ with $0$ referring to the "time variable" ) are the generators of space-time translations (the four-vector $P$ is also called the energy-momentum operator), and one finds that $m \geq 0$ is well-defined by the spectral condition that the (joint) spectrum of $P$ lies in the forward lightcone [a2]. Furthermore, $s$ stands for the spin associated with the representation of the little group, i.e. the subgroup in $\widetilde{P} _ { + } ^ { \uparrow}$ stabilizing a vector $p$ in the Minkowski space-time with Minkowski inner product $p ^ { 2 } = m ^ { 2 }$ and $p ^ { 0 } > 0$. This representation of the little group is furthermore assumed to be finite dimensional for subspaces associated with one-particle states.

In the massless case $m = 0$, the little group stabilizing $p = ( 1,1,00 )$ is $\operatorname{ISO}( 2 )$ and the finite-dimensional representations of this group are characterized by a number $\sigma \in ( 1 / 2 ) \mathbf{Z}$ called the helicity, which has the physical interpretation of the amount of the internal angular momentum (respectively, "spin" ) directed in the flight direction of the particle. The particle type associated with a massless particle is thus denoted by the pair $[ 0 , \sigma ]$.

Denote the field operators of the theory by $A ( f )$, with $f$ from a suitable space of Schwartz test functions (cf. also Generalized functions, space of). If these field operators connect the vacuum $\Omega$ of the theory with the one-particle states with label $[ 0 , \sigma ]$, i.e. $E_{ [ 0 , \sigma ] } A ( f ) \Omega \neq 0$ ($E _{[ 0 , \sigma ]}$ being the projector associated with the one-particle Hilbert space of $[ 0 , \sigma ]$-states), one can develop a scattering theory for the quantum field $A$, following [a3], [a4]: For a test function $f$, let $A _ { f } ( x ) = A ( f _ { x } )$ with $f _ { x } ( y ) = f ( y - x )$. Take $h _ { t } ( s ) = h ( ( s - t ) / \operatorname { log } | t | ) / \operatorname { log } | t | $, with $h$ a positive test function of compact support with $\int h ( s ) d s = 1$ and set

\begin{equation*} A _ { f } ^ { t } = - 2 \int h _ { t } ( s ) \times \left[ \int _ { S ^ { 2 } } d \omega \times ( \frac { \partial } { \partial x ^ { 0 } } A ) _ { f } ( s , \omega s ) \right] s d s, \end{equation*}

where $S ^ { 2 }$ is the unit sphere in $\mathbf{R} ^ { 3 }$ and $d \omega$ means integration over all unit vectors $\omega$ in $S ^ { 2 }$.

Then it was shown in the above-mentioned references (in the axiomatic framework described in [a2], [a3], [a4]), using the Reeh–Schlieder theorem [a2] together with a kind of Huygens principle and locality, that the ( "adiabatic" ) limit $A ^ { \text{in/out} } ( f ) = \operatorname { lim } _ { t \rightarrow \pm \infty } A _ { f } ^ { t }$ can be defined on a suitable dense domain. The quantum fields $A ^ { \text { in/out } } ( f )$ are by definition the free asymptotic quantum fields associated to the field $A ( f )$. Repeated application of the field operators $A ^ { \text { in/out } } ( f )$ to the vacuum $\Omega$ generates the incoming and outgoing multiple particle states of particle type $[ 0 , \sigma ]$, which define in- and out-Fock spaces (cf. also Fock space). The scalar product of states from the in-Fock space with states from the out-Fock space define the scattering matrix for scattering processes, which only involve massless incoming and outgoing particles.

A number of mathematical problems and physical effects arise in the presence of massless fields, as, for example (see the literature for further details): massless fields are intimately connected with long-range forces in elementary particle physics such as e.g. electro-magnetism, see e.g. [a5]. Among others, this leads to mathematical problems in the theory of superselection sectors associated to the algebra of observables of a quantum field theory involving long-range forces [a6].

In the quantum field theory of gauge fields, massless gauge fields are being coupled to Fermionic currents by the Gauss law, which makes it necessary to introduce an indefinite inner product on the state space underlying the quantum field theory and to single out a subspace of "physical states" with positive norm by a gauge principle [a7], [a8], [a9] (for models partly implementing this, see [a10], [a11]).

If massive particles (cf. also Massive field) interact with massless particles, the massive particle can be accompanied by a "cloud" of infinitely many massless particles with finite total energy, which can "smear out" the mass of the massive particle and give rise to the infraparticle problem [a12], [a13]. In the perturbation theory of quantum fields this effect is assumed to justify the mass renormalization, cf. [a5]. Furthermore, mass-zero particles in perturbation theory cause infrared divergences and the problem of summing up Feynman graphs of all orders with "soft" (i.e. low-energy) massless particles [a5].

The occurrence of so-called Goldstone bosons, which are massless particles, is related to symmetry breaking in quantum field theory; for two different aspects of this phenomenon, see e.g. [a5], Vol. II, [a14].


[a1] E.P. Wigner, "On unitary representations of the inhomogenous Lorentz group" Ann. Math. , 40 (1939) pp. 149
[a2] R.F. Streater, A.S. Wightman, "PCT spin & statistics and all that" , Benjamin (1964) MR1884336 MR1045440 MR0468904 MR0161603 Zbl 1026.81027 Zbl 0704.53058 Zbl 0135.44305
[a3] D. Buchholz, "Collision theory for massless Fermions" Commun. Math. Phys. , 42 (1975) pp. 269 MR0371302
[a4] D. Buchholz, "Collision theory for massless Bosons" Commun. Math. Phys. , 52 (1977) pp. 147 MR0436824
[a5] S. Weinberg, "The quantum theory of fields" , I–II , Cambridge Univ. Press (1995) MR2148468 MR2148467 MR2148466 MR1737296 MR1411911 MR1410064 Zbl 1069.81501 Zbl 0949.81001 Zbl 0885.00020 Zbl 0959.81002
[a6] D. Buchholz, "The physical state space of quantum electrodynamics" Commun. Math. Phys. , 85 (1982) pp. 49 MR0667767 Zbl 0506.46052
[a7] F. Strocchi, A.S. Wightman, "Proof of the charge superselection rule in local, relativistic quantum field theory" J. Math. Phys. , 15 (1974) pp. 2198 MR0359606
[a8] F. Strocchi, "Local and covariant gauge quantum field theories, cluster property, superselection rules and the infrared problem" Phys. Rev. , D17 (1978) pp. 2010 MR0523177
[a9] G. Morchio, F. Strocchi, "Infrared singularities, vacuum structure and pure phases in local quantum field theory" Ann. Inst. H. Poincaré , B33 (1980) pp. 251 MR0601840
[a10] S. Albeverio, H. Gottschalk, J.-L. Wu, "Models of local, relativistic quantum fields with indefinite metric (in all dimensions)" Commun. Math. Phys. , 184 (1997) pp. 509
[a11] S. Albeverio, H. Gottschalk, J.-L. Wu, "Nontrivial scattering amplitudes for some local, relativistic quantum field models with indefinite metric" Phys. Lett. , B 405 (1997) pp. 243 MR1461242
[a12] B. Schroer, "Infrateilchen in der Quantenfeldtheorie" Fortschr. Phys. , 173 (1963) pp. 1527 MR0154600
[a13] D. Buchholz, "On the manifestation of particles" A.N. Sen (ed.) A. Gersten (ed.) , Proc. Beer Sheva Conf. (1993): Math. Phys. Towards the 21st Century , Ben Gurion of the Negev Press (1994)
[a14] D. Buchholz, S. Doplicher, R. Longo, J.E. Roberts, "A new look at Goldstone's theorem" Rev. Math. Phys., Special Issue , 49 (1992) MR1199169
How to Cite This Entry:
Massless field. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S. AlbeverioH. Gottschalk (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article