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Lie Groups, Lie Algebras, and Symmetries
Physics — Particle Physics · free-complex-scalar-field-hamiltonian
Problem From (A.7), the Hamiltonian for a free complex scalar field is
H = ∫ d 3 x ⃗ [ ( ∂ ϕ † ∂ t ) ( ∂ ϕ ∂ t ) + ( ∇ ⃗ ϕ † ) ⋅ ( ∇ ⃗ ϕ ) + m 2 ϕ † ϕ ] . H = \int d^3\vec{x} \left[\left(\frac{\partial\phi^\dagger}{\partial t}\right)\left(\frac{\partial\phi}{\partial t}\right) + (\vec{\nabla}\phi^\dagger)\cdot(\vec{\nabla}\phi) + m^2\phi^\dagger\phi\right]. H = ∫ d 3 x [ ( ∂ t ∂ ϕ † ) ( ∂ t ∂ ϕ ) + ( ∇ ϕ † ) ⋅ ( ∇ ϕ ) + m 2 ϕ † ϕ ] . H = ∫ d 3 p ⃗ ( 2 π ) 3 2 E p E p [ a † ( p ⃗ ) a ( p ⃗ ) + b † ( p ⃗ ) b ( p ⃗ ) ] . H = \int \frac{d^3\vec{p}}{(2\pi)^3 2E_p} E_p\left[a^\dagger(\vec{p})a(\vec{p}) + b^\dagger(\vec{p})b(\vec{p})\right]. H = ∫ ( 2 π ) 3 2 E p d 3 p E p [ a † ( p ) a ( p ) + b † ( p ) b ( p ) ] . H H H Browse the problem library