Fabri-Picasso theorem

The Fabri-Picasso theorem , after Elio Fabri and Luigi Ettore Picasso , is a statement from quantum field theory . The theorem says that the operator norm of the charge operator is infinite if the underlying symmetry group is broken explicitly or spontaneously . It follows from this that the charge operator cannot be viewed as an operator on the state space and accordingly no charge can be defined as an eigenvalue of a state. In this sense, the Fabri-Picasso theorem is one of the so-called no-go theorems in quantum field theory.

One consequence of the Fabri-Picasso theorem is that in the standard model, due to the spontaneous symmetry breaking in the electroweak interaction (within the framework of the Higgs mechanism ), there is an electric charge for the electromagnetic interaction , but no analog charge for the weak interaction .

proof

Suppose there is a Hermitian charge operator operating on the state space. Applied to the state of the quantum vacuum , due to the translational invariance of the vacuum and the fact that the charge operator interchanges with the momentum operator : ${\displaystyle {\hat {Q}}}$ ${\displaystyle |\Omega \rangle }$ ${\displaystyle {\hat {P}}}$

${\displaystyle \langle \Omega |{\hat {Q}}{\hat {Q}}|\Omega \rangle =\int \mathrm {d} ^{3}x\,\langle \Omega |j^{0}(x){\hat {Q}}|\Omega \rangle =\int \mathrm {d} ^{3}x\,\langle \Omega |e^{-\mathrm {i} {\hat {P}}\cdot x}j^{0}(0){\hat {Q}}e^{\mathrm {i} {\hat {P}}\cdot x}|\Omega \rangle =\int \mathrm {d} ^{3}x\,\langle \Omega |j^{0}(0){\hat {Q}}|\Omega \rangle }$

In the latter form, the integrand is manifestly independent of location; for a finite result must therefore hold. Exactly this is only the case for unbroken symmetries. ${\displaystyle {\hat {Q}}|\Omega \rangle =0}$

Application to the standard model

The Higgs mechanism results in a non-zero vacuum expectation value, so

${\displaystyle |\Omega \rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}0\\v\end{pmatrix}}}$

is ( has the dimension of an energy , ). The generators of the symmetry group broken by the Higgs mechanism are the three Pauli matrices for and the identity matrix for . There is exactly one nontrivial linear combination that annihilates the vacuum: . From the definition of weak isospin and weak Hyper charge as the charge of respectively the foregoing that the symmetry breaking exactly remains a valid charge - the electric charge: ${\displaystyle v}$ ${\displaystyle v\approx 246\,{\text{GeV}}}$ ${\displaystyle SU(2)_{L}\otimes U(1)_{Y}}$ ${\displaystyle SU(2)_{L}}$ ${\displaystyle U(1)_{Y}}$ ${\displaystyle ({\tfrac {1}{2}}\sigma _{3}+{\tfrac {1}{2}}I_{2})|\Omega \rangle =0}$ ${\displaystyle T}$ ${\displaystyle Y_{W}}$ ${\displaystyle SU(2)_{L}}$ ${\displaystyle U(1)_{Y}}$

${\displaystyle Q=T_{3}+{\tfrac {1}{2}}Y_{W}}$

literature

• Ian JR Aitchison and Anthony JG Hey: Gauge Theories in Particle Physics, Volume 2: Non-Abelian Gauge Theories: QCD and the Electroweak Theory . 3. Edition. Institute of Physics Publishing, Bristol, Philadelphia 2004, ISBN 0-7503-0950-4 (English).
• Elio Fabri and Luigi Ettore Picasso: Quantum Field Theory and Approximate Symmetries . In: Phys. Rev. Lett. tape 16 , no. 10 , 1966, pp. 408-410 (English).