Nonrelativistic Quantum Mechanics
Nonrelativistic Field Operators ¶ One difference in nonrelativistic quantum mechanics is that the
noninteracting solutions to the equation of motion (Schrödinger equation
in this case) can be numbered using a discrete index, so for example the
momentum q \mathbf{q} q
For bosons the field operators are given by:
ψ ^ ( x ) = ∑ k ψ k ( x ) c k \hat\psi({\mathbf{x}}) = \sum_k \psi_k({\mathbf{x}}) c_k ψ ^ ( x ) = k ∑ ψ k ( x ) c k ψ ^ † ( x ) = ∑ k ψ k ∗ ( x ) c k † \hat\psi^\dag({\mathbf{x}}) = \sum_k \psi_k^*({\mathbf{x}}) c_k^\dag ψ ^ † ( x ) = k ∑ ψ k ∗ ( x ) c k † where the coefficients are the single-particle wave functions.
[ c k , c l † ] = δ k l [c_k, c_l^\dag] = \delta_{kl} [ c k , c l † ] = δ k l [ c k , c l ] = [ c k † , c l † ] = 0 [c_k, c_l] = [c_k^\dag, c_l^\dag] = 0 [ c k , c l ] = [ c k † , c l † ] = 0 so the commutation relations for ψ ^ \hat\psi ψ ^ ψ ^ † \hat\psi^\dagger ψ ^ †
[ ψ ^ ( x ) , ψ ^ † ( y ) ] = δ ( 3 ) ( x − y ) [ \hat\psi({\mathbf{x}}), \hat\psi^\dag({\mathbf{y}}) ] = \delta^{(3)}({\mathbf{x}}-{\mathbf{y}}) [ ψ ^ ( x ) , ψ ^ † ( y )] = δ ( 3 ) ( x − y ) [ ψ ^ ( x ) , ψ ^ ( y ) ] = [ ψ ^ † ( x ) , ψ ^ † ( y ) ] = 0 [ \hat\psi({\mathbf{x}}), \hat\psi({\mathbf{y}}) ] =
[ \hat\psi^\dag({\mathbf{x}}), \hat\psi^\dag({\mathbf{y}}) ] = 0 [ ψ ^ ( x ) , ψ ^ ( y )] = [ ψ ^ † ( x ) , ψ ^ † ( y )] = 0 For fermions:
ψ ^ ( x ) = ∑ k ∑ s = 1 2 ψ k s ( x ) c k \hat\psi({\mathbf{x}}) = \sum_k\sum_{s=1}^2 \psi_k^s({\mathbf{x}}) c_k ψ ^ ( x ) = k ∑ s = 1 ∑ 2 ψ k s ( x ) c k ψ ^ † ( x ) = ∑ k ∑ s = 1 2 ψ k s ∗ ( x ) c k † \hat\psi^\dag({\mathbf{x}}) = \sum_k\sum_{s=1}^2 \psi_k^{s*}({\mathbf{x}}) c_k^\dag ψ ^ † ( x ) = k ∑ s = 1 ∑ 2 ψ k s ∗ ( x ) c k † where
{ c k , c l † } = δ k l \{c_k, c_l^\dag\} = \delta_{kl} { c k , c l † } = δ k l { c k , c l } = { c k † , c l † } = 0 \{c_k, c_l\} = \{c_k^\dag, c_l^\dag\} = 0 { c k , c l } = { c k † , c l † } = 0 so the commutation relations for ψ ^ \hat\psi ψ ^ ψ ^ † \hat\psi^\dagger ψ ^ †
{ ψ ^ ( x ) , ψ ^ † ( y ) } = δ ( 3 ) ( x − y ) \{\hat\psi({\mathbf{x}}), \hat\psi^\dag({\mathbf{y}})\} =\delta^{(3)}({\mathbf{x}}-{\mathbf{y}}) { ψ ^ ( x ) , ψ ^ † ( y )} = δ ( 3 ) ( x − y ) { ψ ^ ( x ) , ψ ^ ( y ) } = { ψ ^ † ( x ) , ψ ^ † ( y ) } = 0 \{\hat\psi({\mathbf{x}}), \hat\psi({\mathbf{y}})\} =
\{\hat\psi^\dag({\mathbf{x}}), \hat\psi^\dag({\mathbf{y}})\} = 0 { ψ ^ ( x ) , ψ ^ ( y )} = { ψ ^ † ( x ) , ψ ^ † ( y )} = 0 The (interacting) Hamiltonian for both bosons and fermions is
i ℏ ∂ t ∣ Ψ ( t ) ⟩ = H ^ ∣ Ψ ( t ) ⟩ i\hbar\partial_t|\Psi(t)\rangle = \hat H|\Psi(t)\rangle i ℏ ∂ t ∣Ψ ( t )⟩ = H ^ ∣Ψ ( t )⟩ H ^ = T ^ + V ^ = ∑ i j c i † ⟨ i ∣ T ∣ j ⟩ c j + 1 2 ∑ i j k l c i † c j † ⟨ i j ∣ V ∣ k l ⟩ c l c k \hat H = \hat T + \hat V = \sum_{ij} c_i^\dag\langle i|T|j \rangle c_j +
\frac{1}{2} \sum_{ijkl} c_i^\dag c_j^\dag\langle ij|V|kl \rangle c_l c_k H ^ = T ^ + V ^ = ij ∑ c i † ⟨ i ∣ T ∣ j ⟩ c j + 2 1 ijk l ∑ c i † c j † ⟨ ij ∣ V ∣ k l ⟩ c l c k Note the ordering of the final two destruction operators c l c k c_l c_k c l c k ⟨ i j ∣ V ∣ k l ⟩ \langle ij|V|kl\rangle ⟨ ij ∣ V ∣ k l ⟩
Nonrelativistic Propagator ¶ Nonrelativistic limits of the propagators are obtained by assuming
∥ p ∥ / m c ≪ 1 \|\mathbf{p}\|/mc \ll 1 ∥ p ∥/ m c ≪ 1 ω \omega ω
p 0 c = E = m c 2 + ω p_0 c = E = mc^2 + \omega p 0 c = E = m c 2 + ω p 2 c 2 + m 2 c 4 = E p = m c 2 + p 2 2 m + O ( 1 m c 2 ( p 2 m ) 2 ) \sqrt{{\mathbf{p}}^2c^2 + m^2c^4} = E_{\mathbf{p}} = mc^2+{{\mathbf{p}}^2\over 2m} +
O\left({1\over mc^2}\left(p^2\over m\right)^2\right) p 2 c 2 + m 2 c 4 = E p = m c 2 + 2 m p 2 + O ( m c 2 1 ( m p 2 ) 2 ) we use them to simplify E 2 − [ E ] ( ) p 2 E^2 - [E]()\mathbf{p}^2 E 2 − [ E ] ( ) p 2 c → ∞ c \to \infty c → ∞
1 c 2 ( E 2 − E p 2 ) = 1 c 2 ( E − E p ) ( E + E p ) = {1\over c^2}(E^2 - E_{\mathbf{p}}^2)
= {1\over c^2}(E - E_{\mathbf{p}})(E + E_{\mathbf{p}})
= c 2 1 ( E 2 − E p 2 ) = c 2 1 ( E − E p ) ( E + E p ) = = 1 c 2 ( ω − p 2 2 m + O ( 1 m c 2 ( p 2 m ) 2 ) ) ( 2 m c 2 + ω + p 2 2 m + O ( 1 m c 2 ( p 2 m ) 2 ) ) = = {1\over c^2}\left(\omega-{{\mathbf{p}}^2\over 2m} +
O\left({1\over mc^2}\left(p^2\over m\right)^2\right)
\right)
\left(2mc^2 + \omega + {{\mathbf{p}}^2\over 2m} +
O\left({1\over mc^2}\left(p^2\over m\right)^2\right)
\right)
= = c 2 1 ( ω − 2 m p 2 + O ( m c 2 1 ( m p 2 ) 2 ) ) ( 2 m c 2 + ω + 2 m p 2 + O ( m c 2 1 ( m p 2 ) 2 ) ) = = 2 m ( ω − p 2 2 m + ω 2 m c 2 ( ω − p 2 2 m ) + O ( 1 m c 2 ( p 2 m ) 2 ) ) → =2m\left(\omega-{{\mathbf{p}}^2\over 2m} + {\omega\over 2mc^2}
\left(\omega-{{\mathbf{p}}^2\over 2m}\right) +
O\left({1\over mc^2}\left(p^2\over m\right)^2\right)
\right)
\to = 2 m ( ω − 2 m p 2 + 2 m c 2 ω ( ω − 2 m p 2 ) + O ( m c 2 1 ( m p 2 ) 2 ) ) → → 2 m ( ω − p 2 2 m ) \to 2m\left(\omega-{{\mathbf{p}}^2\over 2m}\right) → 2 m ( ω − 2 m p 2 ) Where E E E ω \omega ω E p E_{\mathbf{p}} E p
D ~ ( p ) = i p 2 − m 2 c 2 + i ϵ = i p 0 2 − p 2 − m 2 c 2 + i ϵ = i 1 c 2 ( p 0 2 c 2 − p 2 c 2 − m 2 c 4 ) + i ϵ = \tilde D(p) = {i\over p^2-m^2c^2+i\epsilon}
= {i\over p_0^2-{\mathbf{p}}^2-m^2c^2+i\epsilon}
= {i\over {1\over c^2}\left(p_0^2c^2-{\mathbf{p}}^2c^2-m^2c^4\right)+i\epsilon}
= D ~ ( p ) = p 2 − m 2 c 2 + i ϵ i = p 0 2 − p 2 − m 2 c 2 + i ϵ i = c 2 1 ( p 0 2 c 2 − p 2 c 2 − m 2 c 4 ) + i ϵ i = = i 1 c 2 ( E 2 − E p 2 ) + i ϵ → i 2 m ( ω − p 2 2 m ) + i ϵ = = {i\over {1\over c^2}\left(E^2 - E_{\mathbf{p}}^2\right)+i\epsilon}
\to
{i\over 2m\left(\omega - {{\mathbf{p}}^2\over 2m}\right)+i\epsilon}
= = c 2 1 ( E 2 − E p 2 ) + i ϵ i → 2 m ( ω − 2 m p 2 ) + i ϵ i = = 1 2 m i ω − p 2 2 m + i ϵ 2 m = 1 2 m i ω − p 2 2 m + i ϵ ′ =
{1\over2m}{i\over\omega-{{\mathbf{p}}^2\over 2m} + i{\epsilon\over 2m}}
=
{1\over2m}{i\over\omega-{{\mathbf{p}}^2\over 2m} + i\epsilon'} = 2 m 1 ω − 2 m p 2 + i 2 m ϵ i = 2 m 1 ω − 2 m p 2 + i ϵ ′ i As you can see, we are interested in the behavior of the propagator in
the vicinity of its positive frequency pole ω ≈ p 2 2 m \omega\approx{\mathbf{p}^2\over 2m} ω ≈ 2 m p 2
Similarly for fermions (we set c = 1 c=1 c = 1
S ~ ( p ) = i ( / p + m ) p 2 − m 2 + i ϵ = i ( p 0 γ 0 − p j γ j + m ) p 2 − m 2 + i ϵ ≈ 1 2 m i ( p 0 γ 0 − p j γ j + m ) ω − p 2 2 m + i ϵ ′ = \tilde S(p) = {i(\slashed{p}+m)\over p^2-m^2+i\epsilon}
= {i(p^0\gamma_0 - p^j\gamma_j+m)\over p^2-m^2+i\epsilon}
\approx
{1\over2m}{i(p^0\gamma_0 - p^j\gamma_j+m)\over
\omega-{{\mathbf{p}}^2\over 2m} + i\epsilon'}
= S ~ ( p ) = p 2 − m 2 + i ϵ i ( / p + m ) = p 2 − m 2 + i ϵ i ( p 0 γ 0 − p j γ j + m ) ≈ 2 m 1 ω − 2 m p 2 + i ϵ ′ i ( p 0 γ 0 − p j γ j + m ) = = 1 2 m i ( ( ω + m ) γ 0 − p j γ j + m ) ω − p 2 2 m + i ϵ ′ ≈ 1 2 m i ( m γ 0 − p j γ j + m ) ω − p 2 2 m + i ϵ ′ = = {1\over2m}{i((\omega+m)\gamma_0 - p^j\gamma_j+m)\over
\omega-{{\mathbf{p}}^2\over 2m} + i\epsilon'}
\approx
{1\over2m}{i(m\gamma_0 - p^j\gamma_j+m)\over
\omega-{{\mathbf{p}}^2\over 2m} + i\epsilon'}
= = 2 m 1 ω − 2 m p 2 + i ϵ ′ i (( ω + m ) γ 0 − p j γ j + m ) ≈ 2 m 1 ω − 2 m p 2 + i ϵ ′ i ( m γ 0 − p j γ j + m ) = [
= i ( 1 2 ( γ 0 + 1 ) − p j γ j 2 m ) ω − p 2 2 m + i ϵ ′ =
{i\left(\frac{1}{2}(\gamma_0+1) - {p^j\gamma_j\over2m}\right)\over
\omega-{{\mathbf{p}}^2\over 2m} + i\epsilon'} = ω − 2 m p 2 + i ϵ ′ i ( 2 1 ( γ 0 + 1 ) − 2 m p j γ j ) ]{label=“fermion-propagator-approx”}
The first term
1 2 ( γ 0 + 1 ) = ( 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ) \frac{1}{2}(\gamma_0+1) = \begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{pmatrix} 2 1 ( γ 0 + 1 ) = ⎝ ⎛ 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 ⎠ ⎞ selects the two upper components of a given bispinor. The second term
− p j γ j 2 m = ( 0 − p j σ j 2 m p j σ j 2 m 0 ) - {p^j\gamma_j\over2m} = \begin{pmatrix}0 & -{p^j\sigma_j\over2m} \\ {p^j\sigma_j\over2m} & 0\end{pmatrix} − 2 m p j γ j = ( 0 2 m p j σ j − 2 m p j σ j 0 ) mixes the upper and lower components of the bispinor and the
contribution of this term is quadratic in p m {\mathbf{p}\over m} m p fermion-propagator-approx{.interpreted-text role=“eq”} reduces to a
unit matrix (in spin space):
S ~ ( p ) ≈ i ( 1 2 ( γ 0 + 1 ) − p j γ j 2 m ) ω − p 2 2 m + i ϵ ≈ i 1 ω − p 2 2 m + i ϵ = 1 G 0 + ( p , ω ) \tilde S(p) \approx
{i\left(\frac{1}{2}(\gamma_0+1) - {p^j\gamma_j\over2m}\right)\over
\omega-{{\mathbf{p}}^2\over 2m} + i\epsilon}
\approx
{i\mathbf{1}\over \omega-{{\mathbf{p}}^2\over 2m} + i\epsilon}
=\mathbf{1} G_0^+({\mathbf{p}}, \omega) S ~ ( p ) ≈ ω − 2 m p 2 + i ϵ i ( 2 1 ( γ 0 + 1 ) − 2 m p j γ j ) ≈ ω − 2 m p 2 + i ϵ i 1 = 1 G 0 + ( p , ω ) where G 0 + ( p , ω ) G_0^+(\mathbf{p}, \omega) G 0 + ( p , ω )
G 0 + ( x − y ) = i ∫ d 3 p ( 2 π ) 3 ∫ d ω 2 π G 0 + ( p , ω ) e i p ⋅ ( x − y ) e − i ω ( t x − t y ) G_0^+(x-y) =
i \int {\mathrm{d}^3 p\over (2\pi)^3}
\int {\mathrm{d} \omega\over 2\pi}
G_0^+({\mathbf{p}}, \omega)
e^{i{\mathbf{p}}\cdot({\mathbf{x}-y})}
e^{-i\omega(t_x - t_y)} G 0 + ( x − y ) = i ∫ ( 2 π ) 3 d 3 p ∫ 2 π d ω G 0 + ( p , ω ) e i p ⋅ ( x − y ) e − iω ( t x − t y ) (For the other pole p 0 = − s q r t p 2 + m 2 p_0 = -sqrt{\mathbf{p}^2+m^2} p 0 = − s q r t p 2 + m 2 ω = − p 0 − m \omega=-p_0-m ω = − p 0 − m
As shown above, the nonrelativistic free propagator is defined by:
G 0 + ( x − y ) = i ∫ d 3 p ( 2 π ) 3 ∫ d ω 2 π G 0 + ( p , ω ) e i p ⋅ ( x − y ) e − i ω ( t x − t y ) G_0^+(x-y) =
i \int {\mathrm{d}^3 p\over (2\pi)^3}
\int {\mathrm{d} \omega\over 2\pi}
G_0^+({\mathbf{p}}, \omega)
e^{i{\mathbf{p}}\cdot({\mathbf{x}-y})}
e^{-i\omega(t_x - t_y)} G 0 + ( x − y ) = i ∫ ( 2 π ) 3 d 3 p ∫ 2 π d ω G 0 + ( p , ω ) e i p ⋅ ( x − y ) e − iω ( t x − t y ) with:
G 0 + ( p , ω ) = i ω − p 2 2 m + i ϵ G_0^+({\mathbf{p}}, \omega)=
{i\over \omega-{{\mathbf{p}}^2\over 2m} + i\epsilon} G 0 + ( p , ω ) = ω − 2 m p 2 + i ϵ i If we use the energies of the nonineracting particles E k ≡ ϵ k = ℏ 2 k 2 2 m = k 2 2 m E_k \equiv \epsilon_k = {\hbar^2 k^2\over 2m} = {k^2\over 2m} E k ≡ ϵ k = 2 m ℏ 2 k 2 = 2 m k 2
G 0 + ( p , ω ) = i ω − p 2 2 m + i ϵ = i ω − E k + i ϵ G_0^+({\mathbf{p}}, \omega)=
{i\over \omega-{{\mathbf{p}}^2\over 2m} + i\epsilon}
=
{i\over \omega-E_k + i\epsilon} G 0 + ( p , ω ) = ω − 2 m p 2 + i ϵ i = ω − E k + i ϵ i so
G 0 + ( k , ω ) = i ω − E k + i ϵ G_0^+(k, \omega) = {i\over \omega-E_k + i\epsilon} G 0 + ( k , ω ) = ω − E k + i ϵ i using E = ℏ ω E = \hbar\omega E = ℏ ω
G 0 + ( k , E ) = i E − E k + i ϵ G_0^+(k, E) = {i\over E-E_k + i\epsilon} G 0 + ( k , E ) = E − E k + i ϵ i Other equivalent ways of representing the propagator:
G 0 + ( x − y ) = G 0 + ( x , t x , y , t y ) = i ∫ d 3 p d E ( 2 π ℏ ) 4 G 0 + ( p , E ) e i ℏ p ⋅ ( x − y ) e − i ℏ E ( t x − t y ) = G_0^+(x-y)
=
G_0^+({\mathbf{x}}, t_x, {\mathbf{y}}, t_y)
=
i \int {\mathrm{d}^3 p\mathrm{d} E\over (2\pi\hbar)^4}
G_0^+({\mathbf{p}}, E)
e^{{i\over\hbar}{\mathbf{p}}\cdot({\mathbf{x}-y})}
e^{-{i\over\hbar}E(t_x - t_y)}
= G 0 + ( x − y ) = G 0 + ( x , t x , y , t y ) = i ∫ ( 2 π ℏ ) 4 d 3 p d E G 0 + ( p , E ) e ℏ i p ⋅ ( x − y ) e − ℏ i E ( t x − t y ) = = i ∫ d 3 k d ω ( 2 π ) 4 G 0 + ( k , ω ) e i k ⋅ ( x − y ) e − i ω ( t x − t y ) =
i \int {\mathrm{d}^3 k\mathrm{d} \omega\over (2\pi)^4}
G_0^+(k, \omega)
e^{i {\mathbf{k}}\cdot({\mathbf{x}-y})}
e^{-i\omega(t_x - t_y)} = i ∫ ( 2 π ) 4 d 3 k d ω G 0 + ( k , ω ) e i k ⋅ ( x − y ) e − iω ( t x − t y ) Sometimes it’s useful to calculate the mixed representation G 0 + ( k , t ) G_0^+(k, t) G 0 + ( k , t )
G 0 + ( k , t ) = ∫ d ω 2 π e − i ω t G 0 + ( k , ω ) = ∫ d ω 2 π e − i ω t i ω − E k + i ϵ = ⋯ = θ t e − i ( E k − i ϵ ) t G_0^+(k, t) = \int {\mathrm{d}\omega\over2\pi}e^{-i\omega t}G_0^+(k, \omega)
= \int {\mathrm{d}\omega\over2\pi}e^{-i\omega t}
{i\over \omega-E_k + i\epsilon}
=
\cdots
=\theta_t e^{-i (E_k-i\epsilon)t} G 0 + ( k , t ) = ∫ 2 π d ω e − iω t G 0 + ( k , ω ) = ∫ 2 π d ω e − iω t ω − E k + i ϵ i = ⋯ = θ t e − i ( E k − i ϵ ) t (The "⋯ \cdots ⋯ t < 0 t < 0 t < 0 t > 0 t > 0 t > 0 θ t \theta_t θ t
Very often, in practice, one just needs to work with G 0 + ( k , t ) G_0^+(k, t) G 0 + ( k , t ) G 0 + ( k , ω ) G_0^+(k, \omega) G 0 + ( k , ω )
G 0 + ( k , t ) = ∫ − ∞ ∞ d ω 2 π e − i ω t G 0 + ( k , ω ) G_0^+(k, t) = \int_{-\infty}^\infty
{\mathrm{d}\omega\over2\pi}e^{-i\omega t}G_0^+(k, \omega) G 0 + ( k , t ) = ∫ − ∞ ∞ 2 π d ω e − iω t G 0 + ( k , ω ) G 0 + ( k , ω ) = ∫ − ∞ ∞ d t e i ω t G 0 + ( k , t ) G_0^+(k, \omega) = \int_{-\infty}^\infty \mathrm{d} t\, e^{i\omega t}G_0^+(k, t) G 0 + ( k , ω ) = ∫ − ∞ ∞ d t e iω t G 0 + ( k , t ) The relation to the contraction of operators is:
G 0 + ( k , t 2 − t 1 ) = − i θ t 2 − t 1 ⟨ Ψ 0 ∣ c k ( t 2 ) c k † ( t 1 ) ∣ Ψ 0 ⟩ G_0^+({\mathbf{k}}, t_2 - t_1) = -i \theta_{t_2-t_1} \langle \Psi_0|
c_{\mathbf{k}}(t_2) c_{\mathbf{k}}^\dag(t_1)|\Psi_0 \rangle G 0 + ( k , t 2 − t 1 ) = − i θ t 2 − t 1 ⟨ Ψ 0 ∣ c k ( t 2 ) c k † ( t 1 ) ∣ Ψ 0 ⟩ where ∣ Ψ 0 ⟩ |\Psi_0\rangle ∣ Ψ 0 ⟩
c k ( t ) = e i H 0 t c k e − i H 0 t c_{\mathbf{k}}(t) = e^{i H_0 t} c_{\mathbf{k}} e^{-i H_0 t} c k ( t ) = e i H 0 t c k e − i H 0 t so to understand the meaning of G 0 + ( k , t 2 − t 1 ) G_0^+(\mathbf{k}, t_2 - t_1) G 0 + ( k , t 2 − t 1 )
G 0 + ( k , t 2 − t 1 ) = − i θ t 2 − t 1 ⟨ Ψ 0 ∣ c k ( t 2 ) c k † ( t 1 ) ∣ Ψ 0 ⟩ = − i θ t 2 − t 1 ⟨ Ψ 0 ∣ e i H 0 t 2 c k e − i H 0 ( t 2 − t 1 ) c k † e − i H 0 t 1 ∣ Ψ 0 ⟩ = G_0^+({\mathbf{k}}, t_2 - t_1) = -i \theta_{t_2-t_1} \langle \Psi_0|
c_{\mathbf{k}}(t_2) c_{\mathbf{k}}^\dag(t_1)|\Psi_0 \rangle
= -i \theta_{t_2-t_1} \langle \Psi_0|e^{i H_0 t_2}
c_{\mathbf{k}}e^{-i H_0 (t_2-t_1)}c_{\mathbf{k}}^\dag e^{-i H_0 t_1}|\Psi_0 \rangle
= G 0 + ( k , t 2 − t 1 ) = − i θ t 2 − t 1 ⟨ Ψ 0 ∣ c k ( t 2 ) c k † ( t 1 ) ∣ Ψ 0 ⟩ = − i θ t 2 − t 1 ⟨ Ψ 0 ∣ e i H 0 t 2 c k e − i H 0 ( t 2 − t 1 ) c k † e − i H 0 t 1 ∣ Ψ 0 ⟩ = = − i θ t 2 − t 1 ( e − i H 0 t 2 ∣ Ψ 0 ⟩ ) † ( c k e − i H 0 ( t 2 − t 1 ) c k † e − i H 0 t 1 ∣ Ψ 0 ⟩ ) = -i \theta_{t_2-t_1} \left(e^{-i H_0 t_2}|\Psi_0\rangle\right)^\dag
\left(
c_{\mathbf{k}}e^{-i H_0 (t_2-t_1)}c_{\mathbf{k}}^\dag e^{-i H_0 t_1}|\Psi_0\rangle
\right) = − i θ t 2 − t 1 ( e − i H 0 t 2 ∣ Ψ 0 ⟩ ) † ( c k e − i H 0 ( t 2 − t 1 ) c k † e − i H 0 t 1 ∣ Ψ 0 ⟩ ) which describes the probability amplitude of adding a bare particle at
time t 1 t_1 t 1 t 2 t_2 t 2 e − i H 0 t 2 ∣ Ψ 0 ⟩ e^{-i H_0 t_2}|\Psi_0\rangle e − i H 0 t 2 ∣ Ψ 0 ⟩