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A [[Quantum field theory|quantum field theory]] is said to contain massless fields if in the [[Hilbert space|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 [[#References|[a1]]], the one-particle states associated with a particle of type $[ m , s ]$ are given by Hilbert subspaces transforming irreducibly under the [[Unitary representation|unitary representation]] of the covering group of the orthochronous proper Poincaré group (cf. also [[Poincaré group|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 [[#References|[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 $p ^ { 0 } &gt; 0$. 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|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 [[#References|[a3]]], [[#References|[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 [[#References|[a2]]], [[#References|[a3]]], [[#References|[a4]]]), using the Reeh–Schlieder theorem [[#References|[a2]]] together with a kind of [[Huygens principle|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|Fock space]]). The scalar product of states from the in-Fock space with states from the out-Fock space define the [[Scattering matrix|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. [[#References|[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 [[#References|[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 [[#References|[a7]]], [[#References|[a8]]], [[#References|[a9]]] (for models partly implementing this, see [[#References|[a10]]], [[#References|[a11]]]).

If massive particles (cf. also [[Massive field|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 [[#References|[a12]]], [[#References|[a13]]]. In the perturbation theory of quantum fields this effect is assumed to justify the mass renormalization, cf. [[#References|[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 [[#References|[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. [[#References|[a5]]], Vol. II, [[#References|[a14]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top"> E.P. Wigner, "On unitary representations of the inhomogenous Lorentz group" ''Ann. Math.'' , '''40''' (1939) pp. 149</td></tr><tr><td valign="top">[a2]</td> <td valign="top"> R.F. Streater, A.S. Wightman, "PCT spin &amp; statistics and all that<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m130/m130080/m13008047.png"/>" , Benjamin (1964) {{MR|1884336}} {{MR|1045440}} {{MR|0468904}} {{MR|0161603}} {{ZBL|1026.81027}} {{ZBL|0704.53058}} {{ZBL|0135.44305}} </td></tr><tr><td valign="top">[a3]</td> <td valign="top"> D. Buchholz, "Collision theory for massless Fermions" ''Commun. Math. Phys.'' , '''42''' (1975) pp. 269 {{MR|0371302}} {{ZBL|}} </td></tr><tr><td valign="top">[a4]</td> <td valign="top"> D. Buchholz, "Collision theory for massless Bosons" ''Commun. Math. Phys.'' , '''52''' (1977) pp. 147 {{MR|0436824}} {{ZBL|}} </td></tr><tr><td valign="top">[a5]</td> <td valign="top"> S. Weinberg, "The quantum theory of fields" , '''I–II''' , Cambridge Univ. Press (1995) {{MR|2148468}} {{MR|2148467}} {{MR|2148466}} {{MR|1737296}} {{MR|1411911}} {{MR|1410064}} {{ZBL|1069.81501}} {{ZBL|0949.81001}} {{ZBL|0885.00020}} {{ZBL|0959.81002}} </td></tr><tr><td valign="top">[a6]</td> <td valign="top"> D. Buchholz, "The physical state space of quantum electrodynamics" ''Commun. Math. Phys.'' , '''85''' (1982) pp. 49 {{MR|0667767}} {{ZBL|0506.46052}} </td></tr><tr><td valign="top">[a7]</td> <td valign="top"> F. Strocchi, A.S. Wightman, "Proof of the charge superselection rule in local, relativistic quantum field theory" ''J. Math. Phys.'' , '''15''' (1974) pp. 2198 {{MR|0359606}} {{ZBL|}} </td></tr><tr><td valign="top">[a8]</td> <td valign="top"> F. Strocchi, "Local and covariant gauge quantum field theories, cluster property, superselection rules and the infrared problem" ''Phys. Rev.'' , '''D17''' (1978) pp. 2010 {{MR|0523177}} {{ZBL|}} </td></tr><tr><td valign="top">[a9]</td> <td valign="top"> G. Morchio, F. Strocchi, "Infrared singularities, vacuum structure and pure phases in local quantum field theory" ''Ann. Inst. H. Poincaré'' , '''B33''' (1980) pp. 251 {{MR|0601840}} {{ZBL|}} </td></tr><tr><td valign="top">[a10]</td> <td valign="top"> 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</td></tr><tr><td valign="top">[a11]</td> <td valign="top"> 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 {{MR|1461242}} {{ZBL|}} </td></tr><tr><td valign="top">[a12]</td> <td valign="top"> B. Schroer, "Infrateilchen in der Quantenfeldtheorie" ''Fortschr. Phys.'' , '''173''' (1963) pp. 1527 {{MR|0154600}} {{ZBL|}} </td></tr><tr><td valign="top">[a13]</td> <td valign="top"> 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)</td></tr><tr><td valign="top">[a14]</td> <td valign="top"> D. Buchholz, S. Doplicher, R. Longo, J.E. Roberts, "A new look at Goldstone's theorem" ''Rev. Math. Phys., Special Issue'' , '''49''' (1992) {{MR|1199169}} {{ZBL|}} </td></tr></table>