RIKEN Accel. Prog. Rep. 48 (2015)

Ⅱ-6. Particle Physics

Spin operator and entanglement in quantum field theory†
K. Fujikawa,∗1 C.H. Oh,∗2 and C. Zhang∗2Entanglement is studied in the framework of Dyson’s
S-matrix theory in relativistic quantum ﬁeld theory,
which leads to a natural deﬁnition of entangled statesof a particle-antiparticle pair and the spin operator
from a Noether current. As an explicit example,
the decay of a massive pseudo-scalar particle into anelectron-positron pair is analyzed. Two spin operators are extracted from the Noether current. The
Wigner spin operator characterizes spin states at therest frame of each fermion and, although not measurable in the laboratory, gives rise to a straightforward
generalization of the low energy analysis of entanglement to the ultra-relativistic domain. In contrast, if
one adopts a (modiﬁed) Dirac spin operator, the entanglement measured using spin correlation becomesmaximal near the threshold of the decay, while the entanglement is replaced by the classical correlation for
the ultra-relativistic electron-positron pair by analogy
to the case of neutrinos, for which a hidden-variablestype description is possible. Entanglement in this sense
depends on the energy scale involved. Chiral symmetry which is fundamental in particle physics diﬀerentiates the spin angular momentum and the magnetic moment. The use of weak interaction, which can measure
helicity, is suggested in the analysis of entanglement
at high energies instead of a Stern-Gerlach apparatus,
which is known to be useless for the electron. A difference between the electron spin at high energies andthe photon linear polarization is also noted.
We formulate the entanglement in the frameworkof relativistic quantum ﬁeld theory, or more precisely,in the S-matrix theory deﬁned by Dyson.1) In the Smatrix theory, we treat only asymptotic states that
contain particles far apart from each other.
We consider the decay of a very massive pseudoscalar particle P into an electron-positron ∫pair byan interaction Hamiltonian, HI (t) = g d3 x :P (x)ψ̄(x)iγ5 ψ(x) : with a coupling constant g. TheDyson∫ formula for the S-matrix gives the state Ψ =−ig d4 x : P (x)ψ̄(x)iγ5 ψ(x) : |P (⃗0)⟩, where we assume a small g. We then obtain for the ﬁxed momentum direction of the electron,1Ψ(⃗p) ≡ √ [a† (⃗p, s)b† (−⃗p, −s) + a† (⃗p, −s)b† (−⃗p, s)]|0⟩.
2This shows a way to prepare a desired state in theframework of local and causal relativistic ﬁeld theory. All the properties of the asymptotic state are ac†∗1∗2

Condensed from the article in Phys. Rev. D. 90, 025028(2014).
RIKEN Nishina Center
Center for Quantum technologies, National University ofSingapore,counted for in the framework which is consistent withlocality, causality and the uncertainty principle; in particular, it is important to recognize that we integrate
over the entire Minkowski space in deﬁning Ψ; thatis, we have no information about when and where theparticle decayed.
Spin Operator
The conserved angular momentum operator (Noether∫ˆcharge) of the Dirac action is given by J⃗ = d3 x :⃗ + S]ψ(x)⃗ψ † (x)[L:, which is written as∫∑1{ξ(s′ )†⃗σ ξ(s)a† (⃗d3 pp, s′ )a(⃗p, s)
2′s,s+∫−ξ † (−s)⃗σ ξ(−s′ )b† (⃗p, s′ )b(⃗p, s)}()∑⃗ p, s)d3 p{a† (⃗p, s) La(⃗s()⃗ p, s) }.
+b† (⃗p, s) Lb(⃗

The ﬁrst term is called the Wigner spin operator,
which is not directly measured in the laboratory. We
instead deﬁne the (modiﬁed) Dirac spin operatorˆ⃗S(⃗p)∑ 1m1≡
ξ † (s′ )⃗σT ξ(s) + p̂ξ † (s′ )(⃗σ · p̂)ξ(s)]{[2E2′s,s×a† (⃗p, s′ )a(⃗p, s)
1m †1ξ (−s)⃗σT ξ(−s′ ) + p̂ξ † (−s)(⃗σ · p̂)ξ(−s′ )]−[2E2×b† (−⃗p, s′ )b(−⃗p, s)},which is close to what is measured in the laboratory.
A salient feature of this spin is that it appraoches thehelicity state proportional to the momentum directionp̂ for E → ∞. That is, the spin correlation at highenergies becomes similar to the correlation of neutrinos. The spin correlation of two neutrinos, which canhave only two states h = ±, does not deﬁne the entanglement in the conventional sense. That is, thehigh energy electron states behave like classical particles, which show the correlation but not entanglement.
This transition to helicity states may be measured using weak interactions.2)
References
1) F. J. Dyson: Phys. Rev. 75, 1736 (1949).
2) K. Fujikawa and R. Shrock: Phys. Rev. Lett. 45, 963(1980).

- 160 -完全版2014_本文.indd 160

15/10/16 17:50