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❙♣✐♥ ♦♥❡ ♠❛tt❡r ✜❡❧❞s

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛

❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s

◆♦✈❡♠❜❡r ✷✵✶✺

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶ ✴ ✷✻

❙♣✐♥ ♦♥❡ ♠❛tt❡r ✜❡❧❞s

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛

❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s

◆♦✈❡♠❜❡r ✷✵✶✺

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✷✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✷ ✴ ✷✻

❖✉t❧✐♥❡ ♦❢ t❤❡ ❚❛❧❦

✶ ▼♦t✐✈❛t✐♦♥s ✷ ❚❤❡ ❍▲● ❛♥❞ P♦✐♥❝❛ré✳ ✸ ❙♣✐♥ ✶ ❛❧❣❡❜r❛ ❛♥❞ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ✐♥ ♠♦♠❡♥t✉♠

s♣❛❝❡✳

✹ ❉②♥❛♠✐❝s ❛♥❞ ❝♦♥str✐❝t✐♦♥s ✺ ◗✉❛♥t✉♠ ❋✐❡❧❞ ❚❤❡♦r② ✻ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ r❡♠❛r❦s

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✸✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✸ ✴ ✷✻

❖✉t❧✐♥❡ ♦❢ t❤❡ ❚❛❧❦

✶ ▼♦t✐✈❛t✐♦♥s ✷ ❚❤❡ ❍▲● ❛♥❞ P♦✐♥❝❛ré✳ ✸ ❙♣✐♥ ✶ ❛❧❣❡❜r❛ ❛♥❞ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ✐♥ ♠♦♠❡♥t✉♠

s♣❛❝❡✳

✹ ❉②♥❛♠✐❝s ❛♥❞ ❝♦♥str✐❝t✐♦♥s ✺ ◗✉❛♥t✉♠ ❋✐❡❧❞ ❚❤❡♦r② ✻ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ r❡♠❛r❦s

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✸✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✸ ✴ ✷✻

❖✉t❧✐♥❡ ♦❢ t❤❡ ❚❛❧❦

✶ ▼♦t✐✈❛t✐♦♥s ✷ ❚❤❡ ❍▲● ❛♥❞ P♦✐♥❝❛ré✳ ✸ ❙♣✐♥ ✶ ❛❧❣❡❜r❛ ❛♥❞ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ✐♥ ♠♦♠❡♥t✉♠

s♣❛❝❡✳

✹ ❉②♥❛♠✐❝s ❛♥❞ ❝♦♥str✐❝t✐♦♥s ✺ ◗✉❛♥t✉♠ ❋✐❡❧❞ ❚❤❡♦r② ✻ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ r❡♠❛r❦s

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✸✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✸ ✴ ✷✻

❖✉t❧✐♥❡ ♦❢ t❤❡ ❚❛❧❦

✶ ▼♦t✐✈❛t✐♦♥s ✷ ❚❤❡ ❍▲● ❛♥❞ P♦✐♥❝❛ré✳ ✸ ❙♣✐♥ ✶ ❛❧❣❡❜r❛ ❛♥❞ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ✐♥ ♠♦♠❡♥t✉♠

s♣❛❝❡✳

✹ ❉②♥❛♠✐❝s ❛♥❞ ❝♦♥str✐❝t✐♦♥s ✺ ◗✉❛♥t✉♠ ❋✐❡❧❞ ❚❤❡♦r② ✻ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ r❡♠❛r❦s

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✸✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✸ ✴ ✷✻

❖✉t❧✐♥❡ ♦❢ t❤❡ ❚❛❧❦

✶ ▼♦t✐✈❛t✐♦♥s ✷ ❚❤❡ ❍▲● ❛♥❞ P♦✐♥❝❛ré✳ ✸ ❙♣✐♥ ✶ ❛❧❣❡❜r❛ ❛♥❞ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ✐♥ ♠♦♠❡♥t✉♠

s♣❛❝❡✳

✹ ❉②♥❛♠✐❝s ❛♥❞ ❝♦♥str✐❝t✐♦♥s ✺ ◗✉❛♥t✉♠ ❋✐❡❧❞ ❚❤❡♦r② ✻ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ r❡♠❛r❦s

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✸✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✸ ✴ ✷✻

❖✉t❧✐♥❡ ♦❢ t❤❡ ❚❛❧❦

✶ ▼♦t✐✈❛t✐♦♥s ✷ ❚❤❡ ❍▲● ❛♥❞ P♦✐♥❝❛ré✳ ✸ ❙♣✐♥ ✶ ❛❧❣❡❜r❛ ❛♥❞ ❡q✉❛t✐♦♥s ♦❢ ♠♦t✐♦♥ ✐♥ ♠♦♠❡♥t✉♠

s♣❛❝❡✳

✹ ❉②♥❛♠✐❝s ❛♥❞ ❝♦♥str✐❝t✐♦♥s ✺ ◗✉❛♥t✉♠ ❋✐❡❧❞ ❚❤❡♦r② ✻ ❈♦♥❝❧✉s✐♦♥s ❛♥❞ r❡♠❛r❦s

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✸✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✸ ✴ ✷✻

❲❡ ♠❛② ♥❡❡❞ t♦ ❧♦♦❦ ✐♥ ♦t❤❡r ❞✐r❡❝t✐♦♥ t♦ ❡①t❡♥❞ t❤❡ ❙▼

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✹✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✹ ✴ ✷✻

❋✐❡❧❞s tr❛♥s❢♦r♠ ✉♥❞❡r t❤ ❍▲●

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✺✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✺ ✴ ✷✻

❋✐❡❧❞s tr❛♥s❢♦r♠ ✉♥❞❡r t❤ ❍▲●

❚❤❡② ❝❛♥ ❜❡ ✉s❡❞ ✐♥ ❡✛❡❝t✐✈❡ t❤❡♦r✐❡s ♦❢ ❝♦♠♣♦✉♥❞ s②st❡♠s ✭RχPT✱ ❤❛❞r♦♥ ♣❤②s✐❝s✮✳ ❚❤❡② ❝❛♥ ❣✐✈❡ ❛❧t❡r♥❛t✐✈❡ r♦✉t❡s t♦ st✉❞② ❞❛r❦ ♠❛tt❡r✳ P♦ss✐❜❧❡ ❡①t❡♥s✐♦♥s t♦ t❤❡ st❛♥❞❛r❞ ♠♦❞❡❧✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✻✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✻ ✴ ✷✻

❋✐❡❧❞s tr❛♥s❢♦r♠ ✉♥❞❡r t❤ ❍▲●

❚❤❡ P♦✐♥❝❛ré ❛❧❣❡❜r❛ ❤❛s t✇♦ ❛❧❣❡❜r❛✐❝ ✐♥✈❛r✐❛♥ts C✷ = PµPµ C✹ = WµW µ with Wµ = ✶ ✷εµστρMστPρ ❖♥❡ ♣❛rt✐❝❧❡ st❛t❡ s❛t✐s❢② C✷|Ψ = m✷|Ψ C✹|Ψ = −m✷j(j + ✶)|Ψ ✇❤❡r❡ ✇❡ ❝❛❧❧ m t❤❡ ♠❛ss ❛♥❞ j t❤❡ s♣✐♥ ♦❢ Ψ✳ ❚❤❡ q✉❛♥t✉♠ ✜❡❧❞s✱ t❤❡ ❜❛s✐❝ ❡❧❡♠❡♥ts ♦❢ ❛ ◗❋❚ ❛❧❧♦✇ ✉s t♦ ❝❛❧❝✉❧❛t❡ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡s✱ ❛r❡ ❜✉✐❧t ❢r♦♠ ♦♣❡r❛t♦rs t❤❛t ❝r❡❛t❡ ♦r ❞❡str♦② t❤✐s st❛t❡s Ψl = ˆ dΓ

• eipxωl (Γ) a† (Γ) + e−ipxωc

l (Γ) a (Γ)

• t❤❡ ✜❡❧❞ ❝♦❡✣❝✐❡♥ts ω✱ tr❛♥s❢♦r♠ ✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ♦❢ t❤❡ ▲♦r❡♥t③

❣r♦✉♣✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✼✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✼ ✴ ✷✻

❍▲● ❛♥❞ ♣❛r✐t②

❚❤❡ ❍▲● ✐s ❛♥ ❤♦♠♦♠♦r♣❤✐s♠ ♦❢ SU (✷) ⊗ SU (✷) ✳ ❚❤✉s t❤❡ r❡♣r❡s❡♥t❛t✐♦♥s ❝❛♥ ❜❡ ❧❛❜❡❧❡❞ ❜② t✇♦ ❛♥❣✉❧❛r ♠♦♠❡♥t❛ (jA, jB)✳ ❇✉t✱ ✉♥❞❡r ♣❛r✐t② (jA, jB) → (jB, jA)✳ ❚♦ ❤❛✈❡ ❛ st❛t❡ ✇✐t❤ ✇❡❧❧ ❞❡✜♥❡❞ ♣❛r✐t② ✇❡ ♠✉st ❡①t❡♥❞ ♦✉r s♣❛❝❡ t♦ (jA, jB) ⊕ (jB, jA) . ❚❤❡♥ t♦ ❞❡s❝r✐❜❡ ❤✐❣❤ s♣✐♥ ♠❛tt❡r ✜❡❧❞s ✇❡ ❝❤♦♦s❡ jA = j ❛♥❞ jB = ✵.

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✽✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✽ ✴ ✷✻

❈♦✈❛r✐❛♥t ❜❛s✐s

■t ✇❛s ♣r♦✈❡♥ ❜② ❙✳ ●♦♠❡③ ❛♥❞ ▼✳ ◆❛♣s✉❝✐❛❧❡✶ t❤❛t t❤❡ ♣❛r✐t② ❜❛s❡❞ ❝♦✈❛r✐❛♥t ❜❛s✐s ❢♦r ❛ ❣❡♥❡r❛❧(j, ✵) ⊕ (✵, j)❝♦♥t❛✐♥s✿ ❚✇♦ ▲♦r❡♥t③ s❝❛❧❛rs✳ ❙✐① ♦♣❡r❛t♦rs tr❛♥s❢♦r♠✐♥❣ ✐♥ (✶, ✵) ⊕ (✵, ✶) ❢♦r♠✐♥❣ ❛ s❡❝♦♥❞ r❛♥❦ ❛♥t②s✐♠♠❡tr✐❝ t❡♥s♦r✳ ❆ ♣❛✐r ♦❢ s②♠♠❡tr✐❝ tr❛❝❡❧❡ss ♠❛tr✐❝❡s Sµ✶µ✷...µj ❆ s❡r✐❡s ♦❢ ♠❛tr✐① t❡♥s♦r ♦♣❡r❛t♦rs✱ ✇✐❝❤ tr❛♥s❢♦r♠ ✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ (✷, ✵) ⊕ (✵, ✷) , (✸, ✵) ⊕ (✵, ✸) , ... (✷j, ✵) ⊕ (✵, ✷j)✳

✶✶✵✳✶✶✵✸✴P❤②s❘❡✈❉✳✽✽✳✵✾✻✵✶✷ ▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✾✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✾ ✴ ✷✻

❚❤❡ ✶

✷, ✵

• ⊕
• ✵, ✶

✷

• ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ✐♥ ♠♦♠❡♥t✉♠ s♣❛❝❡✿

❉✐r❛❝ ❊q✉❛t✐♦♥

✳ ❆s ❛♥ ❡①❛♠♣❧❡ ❧❡t ✉s t❛❦❡ j = ✶

✷✳ ◆♦✇ t❛❦❡ t❤❡ ♣r♦❥❡❝t✐♦♥ ♦✈❡r ♣❛r✐t②

❡✐❣❡♥st❛t❡s Πu (✵) = ±u (♦) ◆♦✇ ✇❡ ❝❛♥ ❛♣♣❧② ❛ ❜♦♦st t♦ t❤✐s ❡q✉❛t✐♦♥ t♦ ♦❜t❛✐♥

• B (♣) ΠB−✶ (♣) ± ✶
• u (♣) = ✵

✐t t✉r♥s ♦✉t t♦ ❜❡ t❤❛t B (♣) ΠB−✶ (♣) = γµpµ m t❤❡♥ ✇❡ r❡❝♦✈❡r t❤❡ ❉✐r❛❝ ❡q✉❛t✐♦♥ (γµpµ ± m) u (♣) = ✵ ✐♥ ♣r✐♥❝✐♣❧❡ ✇❡ ❝❛♥ ❛♣♣❧② t❤❡ s❛♠❡ ♣r♦❝❡❞✉r❡ ❢♦r ❞✐✛❡r❡♥t s♣✐♥s✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✵✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✵ ✴ ✷✻

❚❤❡ (✶, ✵) ⊕ (✵, ✶)❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ✐♥ ♠♦♠❡♥t✉♠ s♣❛❝❡

✳ ❲❡ ❝❛♥ ❣❡t t❤❡ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ❜② ❜♦♦st✐♥❣ t❤❡ r❡st✲❢r❛♠❡ ♣❛r✐t②✲♣r♦❥❡❝t✐♦♥✱ ❜✉t ♥♦✇ t❤❡ ✜❡❧❞s tr❛♥s❢♦r♠ ✐♥ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ (✶, ✵) ⊕ (✵, ✶)♦❢ t❤❡ ▲●✳ ❚❤✐s ✇✐❧❧ ❣✐✈❡ ✉s Sµνpµpν m✷ ± I

• ψ (♣) = ✵ → Λ±ψ (♣) = ψ (♣) ,

✇❤❡r❡ Sµν ✐s ❛ tr❛❝❡❧❡ss t❡♥s♦r ♦❢ r❛♥❦ ✷ ❛♥❞ Λ± ≡ ±✶ ✷ Sµνpµpν m✷ ± I

• ,

✐s t❤❡ ♣r♦❥❡❝t♦r✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✶✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✶ ✴ ✷✻

❲❡ ❤❛✈❡ t♦ ❜❡ ❝❛r❡❢✉❧ ✇❤❡♥ t❤❡ s②st❡♠ ✐s ♦✉t ♦❢ s❤❡❧❧✳

✳ ❚♦ ❣❡t t❤❡ ♣r♦❥❡❝t♦r ♦✉t ♦❢ t❤❡ ♠❛ss s❤❡❧❧ ✇❡ r❡♣❧❛❝❡ m✷ ❜② p✷ ✶ ✷ Sµνpµpν p✷ ∓ I

• ψ (♣) = ∓ψ (♣) ,

❛♥❞ t♦ ❤❛✈❡ ❛ ❧♦❝❛❧ t❤❡♦r② ✇❡ ♣r♦❥❡❝t ♦✈❡r t❤❡ P♦✐♥❝❛ré ♦r❜✐t p✷ = m✷s♦ ✇❡ ♦❜t❛✐♥✿ ✶ ✷ (Sµνpµpν ∓ ηµνpµpν) ψ (♣) = ∓m✷ψ (♣) , ❛♥❞ ✐❢ ✇❡ ❞❡✜♥❡ ❛ ♥❡✇ ♦♣❡r❛t♦r Σµν ≡ ✶

✷ (Sµν ∓ ηµν) ✇❡ ♦❜t❛✐♥✿

• Σµνpµpν ± m✷

u (♣) = ✵.

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✷✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✷ ✴ ✷✻

❚❤❡ ❙ t❡♥s♦r ❤❛✈❡ s♦♠❡ ✐♥t❡r❡st✐♥❣ ♣r♦♣❡rt✐❡s

✳ Sµν ❢✉❧✜❧❧s s♦♠❡ ❏♦r❞❛♥ ❛❧❣❡❜r❛✱ ✇❤✐❝❤ ✐s ❛♥❛❧♦❣♦✉s t♦ t❤❡ ❈❧✐✛♦r❞ ❛❧❣❡❜r❛ ♦❢ t❤❡ γµ✐♥ ❉✐r❛❝ t❤❡♦r②✿

• Sµν, Sαβ

= ✹ ✸

• ηµαηνβ + ηναηµβ − ✶

✷ηµνηαβ

• −✶

✻

• C µανβ + C µβνα

, t❤❡ t❡♥s♦r C µανβ s❛t✐s✜❡s C µανβ = −C αµνβ = C αµβν ✱ C µανβ = C νβµα ❛♥❞ t❤❡ ❇✐❛♥❝❤✐ ✐❞❡♥t✐t②✳ ❚❤❡ ❝♦♠♠✉t❛t♦r ✐s✱ ♦♥ t❤❡ ♦t❤❡r ❤❛♥❞✿

• Sµν, Sαβ

= −i

• ηµαMνβ + ηναMµβ + ηνβMµα + ηµβMνα

. ■t ✐s ❝❧❡❛r ❢r♦♠ ❤❡r❡ t❤❛t S✷ (♣) ≡ SµνSαβpµpνpαpβ = p✹✱ ❛♥❛❧♦❣♦✉s t♦ γµγνpµpν = p✷ ❢♦r ❉✐r❛❝✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✸✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✸ ✴ ✷✻

❲❡ ❝❤♦♦s❡ t❤❡ ♣❛r✐t② ❜❛s✐s ❢♦r ❙

✳ ❚♦ st✉❞② t❤❡ ❞②♥❛♠✐❝s ♦❢ ♦✉r ❡q✉❛t✐♦♥s ✇❡ ♥❡❡❞ t♦ ✇r✐t❡ t❤❡ Sµν ✐♥ s♦♠❡ s♣❡❝✐✜❝ ❜❛s✐s✱ ❢♦r s✐♠♣❧✐❝✐t② ❛♥❞ ❝❧❛r✐t② ✇❡ ❝❤♦♦s❡ t❤❡ ♣❛r✐t② ❜❛s✐s✿ S✵✵ = Π = I ✵ ✵ −I

• S✵i =

✵ −Ji Ji −I

• ,

Sij = ηij +

• Ji, Jj

✵ ✵ −ηij −

• Ji, Jj
• ,

✇❤❡r❡ Ji = ✶

✷ǫijkMjk ❛r❡ t❤❡ ❝♦♥✈❡♥t✐♦♥❛❧ s♣✐♥ ♦♥❡ ♠❛tr✐❝❡s✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✹✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✹ ✴ ✷✻

❚❤❡ ▲❛❣r❛♥❣✐❛♥ ♦❢ t❤❡ t❤❡♦r②

✳ ❆s ✇❡ ❤❛✈❡ s❡❡♥ ♣r❡✈✐♦✉s❧② t❤❡ ♠♦♠❡♥t✉♠ s♣❛❝❡ ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ✐s

• Σµνpµpν − m✷

u (♣) = ✵, ✐♥ ❝♦♥✜❣✉r❛t✐♦♥ s♣❛❝❡ t❤✐s ✇♦✉❧❞ r❡❛❞

• Σµν∂µ∂ν + m✷

Ψ (x) = ✵, ❢r♦♠ ❤❡r❡ ■t t✉r♥s ♦✉t t❤❛t ✇❡ ❝❛♥ ❣❡t t❤✐s ❡q✉❛t✐♦♥ ♦❢ ♠♦t✐♦♥ ❢r♦♠ L = ∂µ ¯ ΨΣµν∂νΨ − m✷ ¯ ΨΨ.

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✺✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✺ ✴ ✷✻

❯s✐♥❣ t❤❡ ❡①♣❧✐❝✐t r❡♣r❡s❡♥t❛t✐♦♥

✳ ❚♦ ✉s❡ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ S ✐t ✐s ❝♦♥✈❡♥✐❡♥t t♦ ✇r✐t❡ Ψ ❛s Ψ = φ ξ

• ,

ς =

• π

, τ

• ✇❤❡r❡ t❤❡ ❝❛♥♦♥✐❝❛❧ ♠♦♠❡♥t✉♠ ❛r❡

πa = δL δ (∂✵φa) = ∂φ†

a − ✶

✷

• ∂iξ†Ji

a ,

❛♥❞ τa = δL δ (∂✵ξa) = −✶ ✷

• ∂iξ†Ji

a ,

❢♦r♠ ❤❡r❡ ✐t ✐s ❝❧❡❛r t❤❛t ✇❡ ❤❛✈❡ t❤❡ r❡str✐❝t✐♦♥s✿ ρa = τa + ✶ ✷

• ∂iξ†Ji

a

ρ†

a = τ † a + ✶

✷

• Ji∂iξ
• a .

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✻✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✻ ✴ ✷✻

❚❤❡ ♥❡✇ ❍❛♠✐❧t♦♥✐❛♥ ✇✐t❤ ❝♦♥str❛✐♥ts

✳ ◆♦✇✱ ❢♦❧❧♦✇✐♥❣ ❉✐r❛❝✱ t❤❡ t✐♠❡ ❡✈♦❧✉t✐♦♥ ♦❢ t❤❡ s②st❡♠ ✐s ❣✐✈❡♥ ❜② H∗ ❞❡✜♥❡❞ ❛s H∗ = ˆ d✸xH + λaρa + λ†

aρ† a.

❚❤❡ ❍❛♠✐❧t♦♥ ❡q✉❛t✐♦♥s t❤❛t ❛r❡ ♠♦❞✐✜❡❞ ✇✐t❤ t❤✐s ❝❤❛♥❣❡ ♦❢ ❍❛♠✐❧t♦♥✐❛♥ ❛r❡✿ ∂✵ξa = δH∗ δτa = λa ∂✵τa = −δH∗ δξa = ✶ ✷∂i

• πJi

a − ✸

✹

• ∂i∂jξ†JiJj

+ m✷ξ†

a,

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✼✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✼ ✴ ✷✻

❙❡❝♦♥❞❛r② ❝♦♥str❛✐♥ts

✳ ■♥ ♦✉r ♣❛rt✐❝✉❧❛r ❝❛s❡ ✇❡ ❞❡✜♥❡ t❤❡ P♦ss✐♦♥ ❜r❛❝❦❡ts ❛s {A (x) , B (y)} = ˆ d✸①➫ δA (①) δΨa δB (②) δςa − δA (②) δςa δB (①) δΨa

• ,

■t ✐s ✈❡r② ❡❛s② t♦ ♣r♦✈❡ t❤❛t {φa (①) , πb (②)} = δabδ✸ (① − ②) {ξa (①) , τb (②)} = δabδ✸ (① − ②) ❚❤❡ ❝♦♥str❛✐♥ts t❤❛t ✇❡ ♦❜t❛✐♥❡❞ ❜❡❢♦r❡ ♠✉st ❜❡ s❛t✐s✜❡❞ ❛t ❛♥② t✐♠❡ t❤✐s ✐♠♣❧✐❡s t❤❛t ∂oρ(†)

a

=

• ρ(†)

a , H∗

= ✵ ❚❤✐s ✇✐❧❧ ♣r♦❞✉❝❡ s❡❝♦♥❞❛r② ❝♦♥str❛✐♥ts ✐♥ ♦✉r t❤❡♦r② χ(†)

a

= ∂i (πJi)(†)

a

− ✶ ✷

• ∂i∂jξ†JiJj(†)

a

+ m✷ξ(†)

a

= ✵. t❤❡r❡ ❛r❡ ♥♦t ❛♥② ♦t❤❡r s❡❝♦♥❞❛r② ❝♦♥str❛✐♥ts✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✽✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✽ ✴ ✷✻

P♦✐ss♦♥ ❇r❛❝❦❡ts

✳ ❆❝❝♦r❞✐♥❣❧② t♦ ❉✐r❛❝ ✇❡ ♠✉st ❝❛❧❝✉❧❛t❡ t❤❡ ♠❛tr✐① ♦❢ P♦✐ss♦♥ ❜r❛❝❦❡ts ∆ab (①, ②) = {fa (①) , fb (②)} = m✷δ✸ (① − ②)     ✵ ✵ ✵ −✶ ✵ ✵ −✶ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ ✵     ✇❤✐❝❤ ❤❛s ❛♥ ✐♥✈❡rs❡ ✭t❤✐s ✐♠♣❧✐❡s t❤❛t t❤❡ ❝♦♥str❛✐♥ts ❛r❡ s❡❝♦♥❞ ❝❧❛ss✮✿ ∆−✶

ab (②, ③) == ✶

m✷ δ✸ (② − ③)     ✵ ✵ ✵ ✶ ✵ ✵ ✶ ✵ ✵ −✶ ✵ ✵ −✶ ✵ ✵ ✵     ❚♦ ❣♦ ❢r♦♠ ❝❧❛ss✐❝❛❧ t♦ q✉❛♥t✉♠ ♠❡❝❤❛♥✐❝s ✇❡ ♣❡r❢♦r♠ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ {A, B}D → i [A, B] ✇❤❡r❡ {A, B}D = {A, B} − ˆ d✸③d✸② {A, fa (③)} ∆−✶

ab (③, ②) {fb (②) , A} .

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✶✾✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✶✾ ✴ ✷✻

❈♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s ❢♦r t❤❡ t❤❡♦r②

✳ ❚❤❡ P♦ss✐♦♥ ❜r❛❝❦❡ts ❢♦r t❤❡ ✜❡❧❞s ❛♥❞ t❤❡✐r r❡s♣❡❝t✐✈❡ ♠♦♠❡♥t✉♠ ❛r❡ {φa (①) , πb (②)}D =

• ✶ − (❏ · ∇)✷

✷m✷

• ab

δ✸ (① − ②) , {ξa (①) , τb (②)}D = (❏ · ∇)✷

ab

✷m✷ δ✸ (① − ②) {ξa (①) , πb (②)}D = {φa (①) , τb (②)}D = ✵ ❛♥❞ ✐♥ ❛ s♣✐♥♦r✐❛❧ ❧❛♥❣✉❛❣❡ ■❢ ✇❡ ❝❛❧❝✉❧❛t❡ {Ψa (①) , ςb (②)}D =

• Σ✵✵ − (❏ · ∇)✷

✷m✷ S✵✵

• ab

δ✸ (① − ②) , t❤❡♥ t♦ ❣♦ t♦ t❤❡ q✉❛♥t✉♠ t❤❡♦r② ✇❡ ♦♥❧② ♠✉st ✐♥❝❧✉❞❡ ❛ i✳ ❛♥❞ ❡q✉❛t❡ t❤✐s t♦ t❤❡ ❝♦♠♠✉t❛t♦r✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✷✵✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✷✵ ✴ ✷✻

❋♦✉r✐❡r ❊①♣❛♥s✐♦♥

❚❤❡ ✜rst st❡♣ ♦❢ ❝❛♥♦♥✐❝❛❧ q✉❛♥t✐③❛t✐♦♥ ✐s t♦ ❡①♣❛♥❞ t❤❡ ✜❡❧❞s ❛s ❛ ❋♦✉r✐❡r s❡r✐❡s✿ Ψ(x) =

• ♣,r

α(♣)

• cr(♣)ur(♣)e−ipx + d+

r (♣)uc r (♣)eipx

❦♥♦✇ ✇❡ ❝❛❧❝✉❧❛t❡ ❛❧❧ t❤❡ ♣❤②s✐❝❛❧ q✉❛♥t✐t✐❡s ❜② ✐♠♣♦s✐♥❣ t❤❡ ✉s✉❛❧ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s t♦ t❤❡ ❝♦❡✣❝✐❡♥ts

• cr (♣) , c†

s (♣)

• = δrsδ♣♣
• dr (♣) , d†

s (♣)

• = δrsδ♣♣

♥♦✇ ✇❡ ❝❛♥ ❝❛❧❝✉❧❛t❡ t❤❡ ❝♦♥❥✉❣❛t❡❞ ♠♦♠❡♥t❛ ✇❤✐❝❤ t✉r♥s ♦✉t t♦ ❜❡ ¯ ςd = ∂L ∂ ¯ Ψd,✵ = Σ✵µ

da (∂µΨ)a

ςd = ∂L ∂Ψd,✵ =

• ∂µ ¯

Ψ

• a Σ✵µ

ad

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✷✶✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✷✶ ✴ ✷✻

❈♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s

❯s✐♥❣ t❤❡ ♦♥ s❤❡❧❧ ♣r♦❥❡❝t♦r ✇❡ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ r❡s✉❧t ❢♦r t❤❡ ❡q✉❛❧ t✐♠❡ ❝♦♠♠✉t❛t✐♦♥ r❡❧❛t✐♦♥s [ςd (①✶) , Ψb (①✷)]x✵

✶✷=✵ = −i

• ♣

pµ ✷Vp✵ Λ (♣)ba Σµ✵

ad

• eipi(xi

✶−xi ✷) − eipi(xi ✶−xi ✷)

♥♦✇ ❝❤❛♥❣✐♥❣ ♣ → −♣ ✐♥ t❤❡ s❡❝♦♥❞ t❡r♠ ❛♥❞ ✉s✐♥❣ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡ S t❡♥s♦r ✇❡ ❣❡t [ςd (①✶) , Ψb (①✷)]x✵

✶✷=✵ = −i

• ♣

eipi(xi

✶−xi ✷)

V

• Σ✵✵ +
• Sij + gijS✵✵

✹m✷ pipj

• ♠❛❦✐♥❣ ✉s❡ ❛❣❛✐♥ ♦❢ t❤❡ ❛❧❣❡❜r❛ ✇❡ ❣❡t ✜♥❛❧❧②

[ςd (①✶) , Ψb (①✷)]x✵

✶✷=✵ = −i

• ♣
• Σ✵✵ + (❏ · ♣)✷ S✵✵

✷m✷

• eipi(xi

✶−xi ✷)

V [ςd (①✶) , Ψb (①✷)]x✵

✶✷=✵ = −i

• Σ✵✵ − (❏ · ∇)✷ S✵✵

✷m✷

• δ✸ (①✶ − ①✷)

t❤✐s ✐s ❥✉st t❤❡ ❡①♣❡❝t❡❞ r❡s✉❧t✳✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✷✷✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✷✷ ✴ ✷✻

❊♥❡r❣②✲♠♦♠❡♥t✉♠ ❛♥❞ ❝❤❛r❣❡ ♦❢ t❤❡ ✜❡❧❞

❚❤❡ ❡♥❡r❣② ♠♦♠❡♥t✉♠ t❡♥s♦r ❛♥❞ ❝✉rr❡♥t ❛r❡ ♦❜t❛✐♥❡❞ ❛s ✉s✉❛❧ T µ

ν = ∂ν ¯

ΨΣµα∂αΨ + ∂α ¯ ΨΣαµ∂νΨ − ηµ

ν

• ∂α ¯

ΨΣαν∂αΨ − m✷ ¯ ΨΨ

• Jα = iq
• (∂µ ¯

Ψ)SµαΨ − ¯ ΨSαν(∂νΨ)

• ❇② s✉❜st✐t✉t✐♥❣ t❤❡ ❋♦✉r✐❡r ❡①♣❛♥s✐♦♥ ✐♥ t❤✐s ❡①♣r❡ss✐♦♥ ✇❡ ❤❛✈❡ ♣r♦✈❡❞ t❤❛t

Pµ =

• ♣,r

[c+

r (♣)cr(♣) + d+ r (♣)dr(♣)]pµ

Q = q

• ♣,r
• d+

r (♣)dr(♣) − c+ r (♣)cr(♣)

• ✇❤✐❝❤ ✐s t❤❡ ❡①♣❡❝t❡❞ r❡s✉❧t ❢♦r♠ ❛ ✇❡❧❧ ❜❡❤❛✈❡❞ t❤❡♦r②✳ ■t ✐s ✐♠♣♦rt❛♥t t♦

r❡♠❛r❦ t❤❛t s♦♠❡ ❢❛❝t♦rs ❛r❡ ♦♥❧② r❡❞✉❝❡❞ ✉s✐♥❣ t❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡ ❙ t❡♥s♦r✳

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✷✸✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✷✸ ✴ ✷✻

✷✲♣♦✐♥t ●r❡❡♥ ❋✉♥❝t✐♦♥

❚❤❡ t✇♦ ♣♦✐♥t ●r❡❡♥ ❋✉♥❝t✐♦♥ iΓF(x − y)ab ✐s t❤❡ t✐♠❡ ♦r❞❡r❡❞ ✈❛❝✉✉♠ ❡①♣❡❝t❛t✐♦♥ ✈❛❧✉❡ ♦❢ t❤❡ ✜❡❧❞s ❛t ❞✐✛❡r❡♥t s♣❛❝❡t✐♠❡ ♣♦✐♥ts✳ ΓF(x − y)ab ≡ ✵|T

• φa(x)¯

φb(y)

• |✵

❢♦r ♦✉r ✜❡❧❞s ✇❡ ❤❛✈❡ ✇❡ ❤❛✈❡ iΓF(x − y)ab =

♣ ✶ ✷V ω♣ Λ (♣)ab e−ipi(xi

✶−xi ✷)

x✵ > y✵

• ♣

✶ ✷V ω♣ Λ (♣)ab eipi(xi

✶−xi ✷)

y✵ > x✵

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✷✹✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✷✹ ✴ ✷✻

✷✲♣♦✐♥t ●r❡❡♥ ❋✉♥❝t✐♦♥

❆❢t❡r ❣♦✐♥❣ t♦ t❤❡ ❝♦♠♣❧❡① ♣❧❛♥❡ ✇❡ ❣❡t t❤❛t t❤❡ ♣r♦♣❛❣❛t♦r ♦❜t❛✐♥❡❞ ❢r♦♠ q✉❛♥t✉♠ ✜❡❧❞ t❤❡♦r② ✐s ✿ iΓF(x − y) = i (✷π)✹ ˆ S (k) + m✷ −

• p✷ − m✷

e−ik(x−y)d✹k ✷m✷ (k✷ − m✷ + iε) +

• S✵✵ − ✶
• δ✹ (x − y)

✷m✷ t❤❡ ❧❛st t❡r♠ ✐s ❛ ❝♦♥t❛❝t t❡r♠✳ ❆❝❝♦r❞✐♥❣❧② t♦ ❲❡✐♥❜❡r❣ t❤❡ ❝♦rr❡❝t ❋❡②♥♠❛♥ r✉❧❡s ❛r❡ ♦❜t❛✐♥❡❞ ❜② ❡❧✐♠✐♥❛t✐♥❣ t❤✐s t❡r♠✳ ❚❤❡♥ iΓF(x − y) = i (✷π)✹ ˆ S (k) + m✷ −

• p✷ − m✷

e−ik(x−y)d✹k ✷m✷ (k✷ − m✷ + iε)

▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✷✺✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✷✺ ✴ ✷✻

❈♦♥❝❧✉s✐♦♥s ❛♥❞ r❡♠❛r❦s

✳

✶ ❚❤❡ ❞✐r❛❝ ❢♦r♠❛❧✐s♠ ❝❛♥ ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❛ ♣r♦❥❡❝t✐♦♥ ♦♥ t♦ ♣❛r✐t②

❡✐❣❡♥st❛t❡s ✐♥ (j, ✵) ⊕ (✵, j)

✷ ❲❡ ❤❛✈❡ ❣❡♥❡r❛❧✐③❡❞ t❤✐s t♦ j = ✶ ❜❛s❡❞ ♦♥ t❤❡ ♣❛r✐t② ❜❛s❡❞ ❝♦✈❛r✐❛♥t

❜❛s✐s ❝♦♥str✉❝t✐♦♥✳

✸ ❚❤❡ ❢♦r♠❛❧✐s♠ ②✐❡❧❞s ❛ ❝♦♥str❛✐♥t ❞②♥❛♠✐❝s✱ ❛❧❧ ❝♦♥str❛✐♥ts ❜❡✐♥❣

s❡❝♦♥❞ ❝❧❛ss✳

✹ ❲❡ ♣❡r❢♦r♠❡❞ t❤❡ ❝❛♥♦♥✐❝❛❧ q✉❛♥t✐③❛t✐♦♥ ❢♦❧❧♦✇✐♥❣ ❉✐r❛❝➫s ❣✉✐❞❡❧✐♥❡s✳ ✺ ❚❤❡ ❛❧❣❡❜r❛ ♦❢ t❤❡ ❙ t❡♥s♦r ✐s ❢✉♥❞❛♠❡♥t❛❧ ❢♦r t❤❡ ❝❛❧❝✉❧❛t✐♦♥s ✐♥ ◗❋❚✳ ✻ ❚❤❡ ❝♦♠♠✉t❛t♦r ♦❢ t❤❡ ✜❡❧❞s ❣✐✈❡s ❛ ♥♦♥ ❝♦♥✈❡♥t✐♦♥❛❧ r❡s✉❧t t❤❛t

❝♦♠❡s ❢r♦♠ t❤❡ ❝♦♥str❛✐♥ts ♦❢ t❤❡ t❤❡♦r②✳

✼ ❚❤❡ ♣r♦♣❛❣❛t♦r ✐♥✈♦❧✈❡s ♥♦t ♦♥❧② t❤❡ ♦♥ s❤❡❧❧ ♣♦❧❛r✐③❛t✐♦♥ s✉♠✱ ❜✉t

❛❧s♦ ✐♥✈♦❧✈❡s t❡r♠s ♣r♦♣♦rt✐♦♥❛❧ t♦ p✷ − m✷✳

✽ P♦ss✐❜❧❡ ❡①t❡♥s✐♦♥s ❛♥❞ ❛♣♣❧✐❝❛t✐♦♥s ♦♥❣♦✐♥❣✳✳✳ ▼✳ ◆❛♣s✉❝✐❛❧❡✱ ❙✳ ❘♦❞r✐❣✉❡③✱ ❘✳❋❡rr♦✲❍❡r♥á♥❞❡③✱ ❙✳ ●♦♠❡③✲➪✈✐❧❛ ✭❯♥✐✈❡rs✐❞❛❞ ❞❡ ●✉❛♥❛❥✉❛t♦ ▼❡①✐❝❛♥ ❲♦r❦s❤♦♣ ♦♥ P❛rt✐❝❧❡s ❛♥❞ ❋✐❡❧❞s✮ ❙♣✐♥❡ ♦♥❡ ✜❡❧❞s✭s❧✐❞❡ ✷✻✮ ◆♦✈❡♠❜❡r ✷✵✶✺ ✷✻ ✴ ✷✻