Propagators

consider the commutators

13.1 The Real Klein–Gordon Field 281

The vacuum |0! E0 = 0

single-particle states a†k |0! Ek = !k

two-particle states a†k1

a†k2

|0! for arbitrary k1 "= k2 Ek1,k2

= !k1 + !k2

1#2

!a†k

"2|0! for arbitrary k Ek,k = 2!k

One obtains a general two-particle state by a linear superposition of thesestates. As a result of (13.1.7), we have a†

k1a†k2

|0! = a†k2

a†k1

|0! . The parti-cles described by the Klein–Gordon field are bosons: each of the occupationnumbers takes the values nk = 0, 1, 2, . . . . The operator nk = a†

kak is theparticle-number operator for particles with the wave vector k whose eigen-values are the occupation numbers nk.

We now turn to the angular momentum of the scalar field. This single-component field contains no intrinsic degrees of freedom and the coe!cientsSrs in Eq. (12.4.25) vanish, Srs = 0. The angular momentum operator(12.4.25) therefore contains no spin component; it comprises only orbitalangular momentum

J =#

d3xx $ P(x) (13.1.20)

= :#

d3xx $ "(x)1i!"(x) : .

The spin of the particles is thus zero. Since the Lagrangian density (13.1.1)and the Hamiltonian (13.1.9) are not gauge invariant, there is no chargeoperator. The real Klein–Gordon field can only describe uncharged particles.An example of a neutral meson with zero spin is the #0.

13.1.2 Propagators

For perturbation theory, and also for the spin-statistic theorem to be dis-cussed later, one requires the vacuum expectation values of bilinear combi-nations of the field operators. To calculate these, we first consider the com-mutators

$"+(x), "+(y)

%=

$"!(x), "!(y)

%= 0

$"+(x), "!(y)

%=

12V

&

k

&

k!

1(!k!k!)1/2

'ak, a†

k!

(e!ikx+ik!y

=12

#d3k

(2#)3e!ik(x!y)

!k, k0 = !k . (13.1.21)

282 13. Free Fields

Using the definitions

!±(x) = ! i2

!d3k

(2")3e!ikx

#k, k0 = #k (13.1.22a)

!(x) =12i

!d3k

(2")31#k

"e"ikx " eikx

#, k0 = #k (13.1.22b)

one can represent the commutators as follows:$$+(x), $"(y)

%= i !+(x " y) (13.1.23a)

$$"(x), $+(y)

%= i !"(x " y) = "i !+(y " x) (13.1.23b)

[$(x), $(y)] =$$+(x), $"(y)

%+

$$"(x), $+(y)

%(13.1.23c)

= i !(x " y) .

We also have the obvious relations

!(x " y) = !+(x " y) + !"(x " y)) (13.1.24a)!"(x) = "!+("x) . (13.1.24b)

In order to emphasize the relativistic covariance of the commutators of thefield, it is convenient to introduce the following four-dimensional integralrepresentations:

!±(x) = "!

C±

d4k

(2")4e"ikx

k2 " m2(13.1.25a)

!(x) = "!

C

d4k

(2")4e"ikx

k2 " m2, (13.1.25b)

for which the contours of integration in the complex k0 plane are shown inFig. 13.1.

Fig. 13.1. Contours of integra-tion C± and C in the com-plex k0 plane for the propaga-tors !±(x) and !(x)

The expressions (13.1.25a,b) can be verified by evaluating the path inte-grals in the complex k0 plane using the residue theorem. The integrands areproportional to [(k0 " #k)(k0 + #k)]"1 and have poles at the positions ±#k.

282 13. Free Fields

Using the definitions

!±(x) = ! i2

!d3k

(2")3e!ikx

#k, k0 = #k (13.1.22a)

!(x) =12i

!d3k

(2")31#k

"e"ikx " eikx

#, k0 = #k (13.1.22b)

one can represent the commutators as follows:$$+(x), $"(y)

%= i !+(x " y) (13.1.23a)

$$"(x), $+(y)

%= i !"(x " y) = "i !+(y " x) (13.1.23b)

[$(x), $(y)] =$$+(x), $"(y)

%+

$$"(x), $+(y)

%(13.1.23c)

= i !(x " y) .

We also have the obvious relations

!(x " y) = !+(x " y) + !"(x " y)) (13.1.24a)!"(x) = "!+("x) . (13.1.24b)

In order to emphasize the relativistic covariance of the commutators of thefield, it is convenient to introduce the following four-dimensional integralrepresentations:

!±(x) = "!

C±

d4k

(2")4e"ikx

k2 " m2(13.1.25a)

!(x) = "!

C

d4k

(2")4e"ikx

k2 " m2, (13.1.25b)

for which the contours of integration in the complex k0 plane are shown inFig. 13.1.

Fig. 13.1. Contours of integra-tion C± and C in the com-plex k0 plane for the propaga-tors !±(x) and !(x)

The expressions (13.1.25a,b) can be verified by evaluating the path inte-grals in the complex k0 plane using the residue theorem. The integrands areproportional to [(k0 " #k)(k0 + #k)]"1 and have poles at the positions ±#k.

282 13. Free Fields

Using the definitions

!±(x) = ! i2

!d3k

(2")3e!ikx

#k, k0 = #k (13.1.22a)

!(x) =12i

!d3k

(2")31#k

"e"ikx " eikx

#, k0 = #k (13.1.22b)

one can represent the commutators as follows:$$+(x), $"(y)

%= i !+(x " y) (13.1.23a)

$$"(x), $+(y)

%= i !"(x " y) = "i !+(y " x) (13.1.23b)

[$(x), $(y)] =$$+(x), $"(y)

%+

$$"(x), $+(y)

%(13.1.23c)

= i !(x " y) .

We also have the obvious relations

!(x " y) = !+(x " y) + !"(x " y)) (13.1.24a)!"(x) = "!+("x) . (13.1.24b)

In order to emphasize the relativistic covariance of the commutators of thefield, it is convenient to introduce the following four-dimensional integralrepresentations:

!±(x) = "!

C±

d4k

(2")4e"ikx

k2 " m2(13.1.25a)

!(x) = "!

C

d4k

(2")4e"ikx

k2 " m2, (13.1.25b)

for which the contours of integration in the complex k0 plane are shown inFig. 13.1.

Fig. 13.1. Contours of integra-tion C± and C in the com-plex k0 plane for the propaga-tors !±(x) and !(x)

The expressions (13.1.25a,b) can be verified by evaluating the path inte-grals in the complex k0 plane using the residue theorem. The integrands areproportional to [(k0 " #k)(k0 + #k)]"1 and have poles at the positions ±#k.

282 13. Free Fields

Using the definitions

!±(x) = ! i2

!d3k

(2")3e!ikx

#k, k0 = #k (13.1.22a)

!(x) =12i

!d3k

(2")31#k

"e"ikx " eikx

#, k0 = #k (13.1.22b)

one can represent the commutators as follows:$$+(x), $"(y)

%= i !+(x " y) (13.1.23a)

$$"(x), $+(y)

%= i !"(x " y) = "i !+(y " x) (13.1.23b)

[$(x), $(y)] =$$+(x), $"(y)

%+

$$"(x), $+(y)

%(13.1.23c)

= i !(x " y) .

We also have the obvious relations

!(x " y) = !+(x " y) + !"(x " y)) (13.1.24a)!"(x) = "!+("x) . (13.1.24b)

In order to emphasize the relativistic covariance of the commutators of thefield, it is convenient to introduce the following four-dimensional integralrepresentations:

!±(x) = "!

C±

d4k

(2")4e"ikx

k2 " m2(13.1.25a)

!(x) = "!

C

d4k

(2")4e"ikx

k2 " m2, (13.1.25b)

for which the contours of integration in the complex k0 plane are shown inFig. 13.1.

Fig. 13.1. Contours of integra-tion C± and C in the com-plex k0 plane for the propaga-tors !±(x) and !(x)

The expressions (13.1.25a,b) can be verified by evaluating the path inte-grals in the complex k0 plane using the residue theorem. The integrands areproportional to [(k0 " #k)(k0 + #k)]"1 and have poles at the positions ±#k.

contours of integration in the complex k0 plane

282 13. Free Fields

Using the definitions

!±(x) = ! i2

!d3k

(2")3e!ikx

#k, k0 = #k (13.1.22a)

!(x) =12i

!d3k

(2")31#k

"e"ikx " eikx

#, k0 = #k (13.1.22b)

one can represent the commutators as follows:$$+(x), $"(y)

%= i !+(x " y) (13.1.23a)

$$"(x), $+(y)

%= i !"(x " y) = "i !+(y " x) (13.1.23b)

[$(x), $(y)] =$$+(x), $"(y)

%+

$$"(x), $+(y)

%(13.1.23c)

= i !(x " y) .

We also have the obvious relations

!(x " y) = !+(x " y) + !"(x " y)) (13.1.24a)!"(x) = "!+("x) . (13.1.24b)

In order to emphasize the relativistic covariance of the commutators of thefield, it is convenient to introduce the following four-dimensional integralrepresentations:

!±(x) = "!

C±

d4k

(2")4e"ikx

k2 " m2(13.1.25a)

!(x) = "!

C

d4k

(2")4e"ikx

k2 " m2, (13.1.25b)

for which the contours of integration in the complex k0 plane are shown inFig. 13.1.

Fig. 13.1. Contours of integra-tion C± and C in the com-plex k0 plane for the propaga-tors !±(x) and !(x)

The expressions (13.1.25a,b) can be verified by evaluating the path inte-grals in the complex k0 plane using the residue theorem. The integrands areproportional to [(k0 " #k)(k0 + #k)]"1 and have poles at the positions ±#k.

residue theorem

The integrands are proportional to [(k0 − ωk)(k0 + ωk)]−1 and have poles at the positions ±ωk

Contour integrals are Lorentz invariant: integrand clearly; integration contour and measure has been shown earlier

vacuum expectation value

13.1 The Real Klein–Gordon Field 283

Depending on the integration path, these poles may contribute to the inte-grals. The right-hand sides of (13.1.25a,b) are manifestly Lorentz covariant.This was shown in (10.1.2) for the volume element, and for the integrand isself-evident.

We now turn to the evaluation of the vacuum expectation values andpropagators. Taking the vacuum expectation value of (13.1.23a) and using! |0! = 0, one obtains

i "+(x " x!) = #0| [!+(x), !"(x!)] |0! = #0|!+(x)!"(x!) |0!= #0|!(x)!(x!) |0! . (13.1.26)

In perturbation theory (Sect. 15.2) we will encounter time-ordered productsof the perturbation Hamiltonian. For their evaluation we will need vacuumexpectation values of time-ordered products. The time-ordered product T isdefined for bosons as follows:

T !(x)!(x!) =!

!(x)!(x!) t > t!

!(x!)!(x) t < t!(13.1.27)

= #(t " t!)!(x)!(x!) + #(t! " t)!(x!)!(x) .

Fig. 13.2. Contour of integration in thek0 plane for the Feynman propagator!F(x)

The Feynman propagator is defined in terms of the expectation value ofthe time-ordered product:

i "F(x " x!) $ #0|T (!(x)!(x!)) |0! (13.1.28)= i (#(t " t!)"+(x " x!) " #(t! " t)""(x " x!)) .

This is related to "±(x) through

"F(x) = ±"±(x) for t ! 0 (13.1.29)

and has the integral representation

"F(x) ="

CF

d4k

(2$)4e"ikx

k2 " m2. (13.1.30)

The latter can be seen by adding an infinite half-circle to the integra-tion contour in the upper or lower half-plane of k0 and comparing it with

using φ |0⟩ = 0

time-ordered product T for bosons

13.1 The Real Klein–Gordon Field 283

Depending on the integration path, these poles may contribute to the inte-grals. The right-hand sides of (13.1.25a,b) are manifestly Lorentz covariant.This was shown in (10.1.2) for the volume element, and for the integrand isself-evident.

We now turn to the evaluation of the vacuum expectation values andpropagators. Taking the vacuum expectation value of (13.1.23a) and using! |0! = 0, one obtains

i "+(x " x!) = #0| [!+(x), !"(x!)] |0! = #0|!+(x)!"(x!) |0!= #0|!(x)!(x!) |0! . (13.1.26)

In perturbation theory (Sect. 15.2) we will encounter time-ordered productsof the perturbation Hamiltonian. For their evaluation we will need vacuumexpectation values of time-ordered products. The time-ordered product T isdefined for bosons as follows:

T !(x)!(x!) =!

!(x)!(x!) t > t!

!(x!)!(x) t < t!(13.1.27)

= #(t " t!)!(x)!(x!) + #(t! " t)!(x!)!(x) .

Fig. 13.2. Contour of integration in thek0 plane for the Feynman propagator!F(x)

The Feynman propagator is defined in terms of the expectation value ofthe time-ordered product:

i "F(x " x!) $ #0|T (!(x)!(x!)) |0! (13.1.28)= i (#(t " t!)"+(x " x!) " #(t! " t)""(x " x!)) .

This is related to "±(x) through

"F(x) = ±"±(x) for t ! 0 (13.1.29)

and has the integral representation

"F(x) ="

CF

d4k

(2$)4e"ikx

k2 " m2. (13.1.30)

The latter can be seen by adding an infinite half-circle to the integra-tion contour in the upper or lower half-plane of k0 and comparing it with

The Feynman propagator is defined in terms of the expectation value of the time-ordered product:

13.1 The Real Klein–Gordon Field 283

Depending on the integration path, these poles may contribute to the inte-grals. The right-hand sides of (13.1.25a,b) are manifestly Lorentz covariant.This was shown in (10.1.2) for the volume element, and for the integrand isself-evident.

We now turn to the evaluation of the vacuum expectation values andpropagators. Taking the vacuum expectation value of (13.1.23a) and using! |0! = 0, one obtains

i "+(x " x!) = #0| [!+(x), !"(x!)] |0! = #0|!+(x)!"(x!) |0!= #0|!(x)!(x!) |0! . (13.1.26)

In perturbation theory (Sect. 15.2) we will encounter time-ordered productsof the perturbation Hamiltonian. For their evaluation we will need vacuumexpectation values of time-ordered products. The time-ordered product T isdefined for bosons as follows:

T !(x)!(x!) =!

!(x)!(x!) t > t!

!(x!)!(x) t < t!(13.1.27)

= #(t " t!)!(x)!(x!) + #(t! " t)!(x!)!(x) .

Fig. 13.2. Contour of integration in thek0 plane for the Feynman propagator!F(x)

The Feynman propagator is defined in terms of the expectation value ofthe time-ordered product:

i "F(x " x!) $ #0|T (!(x)!(x!)) |0! (13.1.28)= i (#(t " t!)"+(x " x!) " #(t! " t)""(x " x!)) .

This is related to "±(x) through

"F(x) = ±"±(x) for t ! 0 (13.1.29)

and has the integral representation

"F(x) ="

CF

d4k

(2$)4e"ikx

k2 " m2. (13.1.30)

The latter can be seen by adding an infinite half-circle to the integra-tion contour in the upper or lower half-plane of k0 and comparing it with

13.1 The Real Klein–Gordon Field 283

Depending on the integration path, these poles may contribute to the inte-grals. The right-hand sides of (13.1.25a,b) are manifestly Lorentz covariant.This was shown in (10.1.2) for the volume element, and for the integrand isself-evident.

We now turn to the evaluation of the vacuum expectation values andpropagators. Taking the vacuum expectation value of (13.1.23a) and using! |0! = 0, one obtains

i "+(x " x!) = #0| [!+(x), !"(x!)] |0! = #0|!+(x)!"(x!) |0!= #0|!(x)!(x!) |0! . (13.1.26)

In perturbation theory (Sect. 15.2) we will encounter time-ordered productsof the perturbation Hamiltonian. For their evaluation we will need vacuumexpectation values of time-ordered products. The time-ordered product T isdefined for bosons as follows:

T !(x)!(x!) =!

!(x)!(x!) t > t!

!(x!)!(x) t < t!(13.1.27)

= #(t " t!)!(x)!(x!) + #(t! " t)!(x!)!(x) .

Fig. 13.2. Contour of integration in thek0 plane for the Feynman propagator!F(x)

The Feynman propagator is defined in terms of the expectation value ofthe time-ordered product:

i "F(x " x!) $ #0|T (!(x)!(x!)) |0! (13.1.28)= i (#(t " t!)"+(x " x!) " #(t! " t)""(x " x!)) .

This is related to "±(x) through

"F(x) = ±"±(x) for t ! 0 (13.1.29)

and has the integral representation

"F(x) ="

CF

d4k

(2$)4e"ikx

k2 " m2. (13.1.30)

The latter can be seen by adding an infinite half-circle to the integra-tion contour in the upper or lower half-plane of k0 and comparing it withThe latter can be seen by adding an infinite half-circle to the integration contour in the upper or lower half-plane of k0.The integration along the path CF shown in the figure is identical to the integration along the real k0 axis, whereby the infinitesimal displacements η and ε in the integrands serve to shift the poles away from the real axis:

13.1 The Real Klein–Gordon Field 283

Depending on the integration path, these poles may contribute to the inte-grals. The right-hand sides of (13.1.25a,b) are manifestly Lorentz covariant.This was shown in (10.1.2) for the volume element, and for the integrand isself-evident.

We now turn to the evaluation of the vacuum expectation values andpropagators. Taking the vacuum expectation value of (13.1.23a) and using! |0! = 0, one obtains

i "+(x " x!) = #0| [!+(x), !"(x!)] |0! = #0|!+(x)!"(x!) |0!= #0|!(x)!(x!) |0! . (13.1.26)

In perturbation theory (Sect. 15.2) we will encounter time-ordered productsof the perturbation Hamiltonian. For their evaluation we will need vacuumexpectation values of time-ordered products. The time-ordered product T isdefined for bosons as follows:

T !(x)!(x!) =!

!(x)!(x!) t > t!

!(x!)!(x) t < t!(13.1.27)

= #(t " t!)!(x)!(x!) + #(t! " t)!(x!)!(x) .

Fig. 13.2. Contour of integration in thek0 plane for the Feynman propagator!F(x)

The Feynman propagator is defined in terms of the expectation value ofthe time-ordered product:

i "F(x " x!) $ #0|T (!(x)!(x!)) |0! (13.1.28)= i (#(t " t!)"+(x " x!) " #(t! " t)""(x " x!)) .

This is related to "±(x) through

"F(x) = ±"±(x) for t ! 0 (13.1.29)

and has the integral representation

"F(x) ="

CF

d4k

(2$)4e"ikx

k2 " m2. (13.1.30)

The latter can be seen by adding an infinite half-circle to the integra-tion contour in the upper or lower half-plane of k0 and comparing it with

284 13. Free Fields

Eq.(13.1.25a). The integration along the path CF defined in Fig. 13.2 is iden-tical to the integration along the real k0 axis, whereby the infinitesimal dis-placements ! and " in the integrands serve to shift the poles k0 = ±(#k!i!) =±(

"k2 + m2 ! i!) away from the real axis:

$F(x) = lim!!0+

!d4k

(2%)4e"ikx

k20 ! (#k ! i!)2

(13.1.31)

= lim"!0+

!d4k

(2%)4e"ikx

k2 ! m2 + i".

As a preparation for the perturbation-theoretical representation in terms ofFeynman diagrams, it is useful to give a pictorial description of the processesrepresented by propagators. Plotting the time axis to the right in Fig. 13.3,

b)a)

time time

xx

x!x!

Fig. 13.3. Propagation of a particle (a)from x! to x and (b) from x to x!

diagram (a) means that a meson is created at x# and subsequently annihilatedat x, i.e., it is the process described by #0|&(x)&(x#) |0$ = i$+(x ! x#). Dia-gram (b) represents the creation of a particle at x and its annihilation at x#,i.e., #0|&(x#)&(x) |0$ = !i$"(x ! x#). Both processes together are describedby the Feynman propagator for the mesons of the Klein–Gordon field, whichis thus often called, for short, the meson propagator.

As an example, we consider the scattering of two nucleons, which arerepresented in Fig. 13.4 by the full lines. The scattering arises due to the

=+

xx x

x!x! x!

!F

Fig. 13.4. Graphical representation of the meson propagator !F(x ! x!) . In thefirst diagram, a meson is created at x! and annihilated at x. In the second diagram,a meson is created at x and annihialted at x!. Full lines represent nucleons, anddashed lines mesons

pictorial description

284 13. Free Fields

Eq.(13.1.25a). The integration along the path CF defined in Fig. 13.2 is iden-tical to the integration along the real k0 axis, whereby the infinitesimal dis-placements ! and " in the integrands serve to shift the poles k0 = ±(#k!i!) =±(

"k2 + m2 ! i!) away from the real axis:

$F(x) = lim!!0+

!d4k

(2%)4e"ikx

k20 ! (#k ! i!)2

(13.1.31)

= lim"!0+

!d4k

(2%)4e"ikx

k2 ! m2 + i".

As a preparation for the perturbation-theoretical representation in terms ofFeynman diagrams, it is useful to give a pictorial description of the processesrepresented by propagators. Plotting the time axis to the right in Fig. 13.3,

b)a)

time time

xx

x!x!

Fig. 13.3. Propagation of a particle (a)from x! to x and (b) from x to x!

diagram (a) means that a meson is created at x# and subsequently annihilatedat x, i.e., it is the process described by #0|&(x)&(x#) |0$ = i$+(x ! x#). Dia-gram (b) represents the creation of a particle at x and its annihilation at x#,i.e., #0|&(x#)&(x) |0$ = !i$"(x ! x#). Both processes together are describedby the Feynman propagator for the mesons of the Klein–Gordon field, whichis thus often called, for short, the meson propagator.

As an example, we consider the scattering of two nucleons, which arerepresented in Fig. 13.4 by the full lines. The scattering arises due to the

=+

xx x

x!x! x!

!F

Fig. 13.4. Graphical representation of the meson propagator !F(x ! x!) . In thefirst diagram, a meson is created at x! and annihilated at x. In the second diagram,a meson is created at x and annihialted at x!. Full lines represent nucleons, anddashed lines mesons

Propagation of a particle (a) from x! to x and (b)from x to x!

scattering of two nucleons: scattering arises due to the exchange of mesons

284 13. Free Fields

Eq.(13.1.25a). The integration along the path CF defined in Fig. 13.2 is iden-tical to the integration along the real k0 axis, whereby the infinitesimal dis-placements ! and " in the integrands serve to shift the poles k0 = ±(#k!i!) =±(

"k2 + m2 ! i!) away from the real axis:

$F(x) = lim!!0+

!d4k

(2%)4e"ikx

k20 ! (#k ! i!)2

(13.1.31)

= lim"!0+

!d4k

(2%)4e"ikx

k2 ! m2 + i".

As a preparation for the perturbation-theoretical representation in terms ofFeynman diagrams, it is useful to give a pictorial description of the processesrepresented by propagators. Plotting the time axis to the right in Fig. 13.3,

b)a)

time time

xx

x!x!

Fig. 13.3. Propagation of a particle (a)from x! to x and (b) from x to x!

diagram (a) means that a meson is created at x# and subsequently annihilatedat x, i.e., it is the process described by #0|&(x)&(x#) |0$ = i$+(x ! x#). Dia-gram (b) represents the creation of a particle at x and its annihilation at x#,i.e., #0|&(x#)&(x) |0$ = !i$"(x ! x#). Both processes together are describedby the Feynman propagator for the mesons of the Klein–Gordon field, whichis thus often called, for short, the meson propagator.

As an example, we consider the scattering of two nucleons, which arerepresented in Fig. 13.4 by the full lines. The scattering arises due to the

=+

xx x

x!x! x!

!F

Fig. 13.4. Graphical representation of the meson propagator !F(x ! x!) . In thefirst diagram, a meson is created at x! and annihilated at x. In the second diagram,a meson is created at x and annihialted at x!. Full lines represent nucleons, anddashed lines mesons

The Complex Klein–Gordon Field

13.2 The Complex Klein–Gordon Field 285

exchange of mesons. The two processes are represented jointly, and indepen-dently of their temporal sequence, by the Feynman propagator.

13.2 The Complex Klein–Gordon Field

The complex Klein–Gordon field is very similar to the real Klein–Gordonfield, except that now the particles created and annihilated by the field carrya charge. Our starting point is the Lagrangian density

L = : !†,µ(x)!,µ(x) ! m2!†(x)!(x) : . (13.2.1)

In line with the remark following Eq. (12.2.24), !(x) and !†(x) are treated asindependent fields. Hence, we have, for example, !L

!"†,µ(x)

= !,µ(x), and, fromthe Euler–Lagrange equations (12.2.15), the equations of motion

("µ"µ + m2)!(x) = 0 and ("µ"µ + m2)!†(x) = 0 . (13.2.2)

The conjugate fields of !(x) and !†(x) are, according to (12.2.16),

#(x) = !†(x) and #†(x) = !(x) . (13.2.3)

Since the complex Klein–Gordon field also behaves as a scalar under Lorentztransformations, it has spin = 0. Due to the gauge invariance of L, this fieldpossesses an additional conserved quantity, namely the charge Q. The equaltime commutators of the fields and their adjoints are, according to canonicalquantization (12.3.1),

!!(x, t), !†(x!, t)

"= i $(x ! x!)

!!†(x, t), !(x!, t)

"= i $(x ! x!)

(13.2.4)

and!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"

=!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"= 0 .

The solutions of the field equations (13.2.2) for the complex Klein–Gordonfield are also of the form e±ikx, so that the expansion of the field operatortakes the form

!(x) = !+(x) + !"(x) =#

k

1(2V %k)1/2

$ake"ikx + b†keikx

%, (13.2.5a)

where, in contrast to the real Klein–Gordon field, the amplitudes b†k and ak

are now independent of one another. From (13.2.5a) we have

13.2 The Complex Klein–Gordon Field 285

exchange of mesons. The two processes are represented jointly, and indepen-dently of their temporal sequence, by the Feynman propagator.

13.2 The Complex Klein–Gordon Field

The complex Klein–Gordon field is very similar to the real Klein–Gordonfield, except that now the particles created and annihilated by the field carrya charge. Our starting point is the Lagrangian density

L = : !†,µ(x)!,µ(x) ! m2!†(x)!(x) : . (13.2.1)

In line with the remark following Eq. (12.2.24), !(x) and !†(x) are treated asindependent fields. Hence, we have, for example, !L

!"†,µ(x)

= !,µ(x), and, fromthe Euler–Lagrange equations (12.2.15), the equations of motion

("µ"µ + m2)!(x) = 0 and ("µ"µ + m2)!†(x) = 0 . (13.2.2)

The conjugate fields of !(x) and !†(x) are, according to (12.2.16),

#(x) = !†(x) and #†(x) = !(x) . (13.2.3)

Since the complex Klein–Gordon field also behaves as a scalar under Lorentztransformations, it has spin = 0. Due to the gauge invariance of L, this fieldpossesses an additional conserved quantity, namely the charge Q. The equaltime commutators of the fields and their adjoints are, according to canonicalquantization (12.3.1),

!!(x, t), !†(x!, t)

"= i $(x ! x!)

!!†(x, t), !(x!, t)

"= i $(x ! x!)

(13.2.4)

and!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"

=!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"= 0 .

The solutions of the field equations (13.2.2) for the complex Klein–Gordonfield are also of the form e±ikx, so that the expansion of the field operatortakes the form

!(x) = !+(x) + !"(x) =#

k

1(2V %k)1/2

$ake"ikx + b†keikx

%, (13.2.5a)

where, in contrast to the real Klein–Gordon field, the amplitudes b†k and ak

are now independent of one another. From (13.2.5a) we have

φ(x) and φ†(x) are treated as independent fields

Euler–Lagrange equations

13.2 The Complex Klein–Gordon Field 285

exchange of mesons. The two processes are represented jointly, and indepen-dently of their temporal sequence, by the Feynman propagator.

13.2 The Complex Klein–Gordon Field

The complex Klein–Gordon field is very similar to the real Klein–Gordonfield, except that now the particles created and annihilated by the field carrya charge. Our starting point is the Lagrangian density

L = : !†,µ(x)!,µ(x) ! m2!†(x)!(x) : . (13.2.1)

In line with the remark following Eq. (12.2.24), !(x) and !†(x) are treated asindependent fields. Hence, we have, for example, !L

!"†,µ(x)

= !,µ(x), and, fromthe Euler–Lagrange equations (12.2.15), the equations of motion

("µ"µ + m2)!(x) = 0 and ("µ"µ + m2)!†(x) = 0 . (13.2.2)

The conjugate fields of !(x) and !†(x) are, according to (12.2.16),

#(x) = !†(x) and #†(x) = !(x) . (13.2.3)

Since the complex Klein–Gordon field also behaves as a scalar under Lorentztransformations, it has spin = 0. Due to the gauge invariance of L, this fieldpossesses an additional conserved quantity, namely the charge Q. The equaltime commutators of the fields and their adjoints are, according to canonicalquantization (12.3.1),

!!(x, t), !†(x!, t)

"= i $(x ! x!)

!!†(x, t), !(x!, t)

"= i $(x ! x!)

(13.2.4)

and!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"

=!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"= 0 .

The solutions of the field equations (13.2.2) for the complex Klein–Gordonfield are also of the form e±ikx, so that the expansion of the field operatortakes the form

!(x) = !+(x) + !"(x) =#

k

1(2V %k)1/2

$ake"ikx + b†keikx

%, (13.2.5a)

where, in contrast to the real Klein–Gordon field, the amplitudes b†k and ak

are now independent of one another. From (13.2.5a) we have

13.2 The Complex Klein–Gordon Field 285

exchange of mesons. The two processes are represented jointly, and indepen-dently of their temporal sequence, by the Feynman propagator.

13.2 The Complex Klein–Gordon Field

The complex Klein–Gordon field is very similar to the real Klein–Gordonfield, except that now the particles created and annihilated by the field carrya charge. Our starting point is the Lagrangian density

L = : !†,µ(x)!,µ(x) ! m2!†(x)!(x) : . (13.2.1)

In line with the remark following Eq. (12.2.24), !(x) and !†(x) are treated asindependent fields. Hence, we have, for example, !L

!"†,µ(x)

= !,µ(x), and, fromthe Euler–Lagrange equations (12.2.15), the equations of motion

("µ"µ + m2)!(x) = 0 and ("µ"µ + m2)!†(x) = 0 . (13.2.2)

The conjugate fields of !(x) and !†(x) are, according to (12.2.16),

#(x) = !†(x) and #†(x) = !(x) . (13.2.3)

Since the complex Klein–Gordon field also behaves as a scalar under Lorentztransformations, it has spin = 0. Due to the gauge invariance of L, this fieldpossesses an additional conserved quantity, namely the charge Q. The equaltime commutators of the fields and their adjoints are, according to canonicalquantization (12.3.1),

!!(x, t), !†(x!, t)

"= i $(x ! x!)

!!†(x, t), !(x!, t)

"= i $(x ! x!)

(13.2.4)

and!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"

=!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"= 0 .

The solutions of the field equations (13.2.2) for the complex Klein–Gordonfield are also of the form e±ikx, so that the expansion of the field operatortakes the form

!(x) = !+(x) + !"(x) =#

k

1(2V %k)1/2

$ake"ikx + b†keikx

%, (13.2.5a)

where, in contrast to the real Klein–Gordon field, the amplitudes b†k and ak

are now independent of one another. From (13.2.5a) we have

Since the complex Klein–Gordon field also behaves as a scalar under Lorentz transformations, it has spin = 0. Due to the gauge invariance of L, this field possesses an additional conserved quantity, namely the charge Q. The equal time commutators of the fields and their adjoints are

13.2 The Complex Klein–Gordon Field 285

exchange of mesons. The two processes are represented jointly, and indepen-dently of their temporal sequence, by the Feynman propagator.

13.2 The Complex Klein–Gordon Field

The complex Klein–Gordon field is very similar to the real Klein–Gordonfield, except that now the particles created and annihilated by the field carrya charge. Our starting point is the Lagrangian density

L = : !†,µ(x)!,µ(x) ! m2!†(x)!(x) : . (13.2.1)

In line with the remark following Eq. (12.2.24), !(x) and !†(x) are treated asindependent fields. Hence, we have, for example, !L

!"†,µ(x)

= !,µ(x), and, fromthe Euler–Lagrange equations (12.2.15), the equations of motion

("µ"µ + m2)!(x) = 0 and ("µ"µ + m2)!†(x) = 0 . (13.2.2)

The conjugate fields of !(x) and !†(x) are, according to (12.2.16),

#(x) = !†(x) and #†(x) = !(x) . (13.2.3)

Since the complex Klein–Gordon field also behaves as a scalar under Lorentztransformations, it has spin = 0. Due to the gauge invariance of L, this fieldpossesses an additional conserved quantity, namely the charge Q. The equaltime commutators of the fields and their adjoints are, according to canonicalquantization (12.3.1),

!!(x, t), !†(x!, t)

"= i $(x ! x!)

!!†(x, t), !(x!, t)

"= i $(x ! x!)

(13.2.4)

and!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"

=!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"= 0 .

The solutions of the field equations (13.2.2) for the complex Klein–Gordonfield are also of the form e±ikx, so that the expansion of the field operatortakes the form

!(x) = !+(x) + !"(x) =#

k

1(2V %k)1/2

$ake"ikx + b†keikx

%, (13.2.5a)

where, in contrast to the real Klein–Gordon field, the amplitudes b†k and ak

are now independent of one another. From (13.2.5a) we have

The solutions of the field equations (13.2.2) for the complex Klein–Gordon field are also of the form e±ikx, so that the expansion of the field operator takes the form

13.2 The Complex Klein–Gordon Field 285

exchange of mesons. The two processes are represented jointly, and indepen-dently of their temporal sequence, by the Feynman propagator.

13.2 The Complex Klein–Gordon Field

The complex Klein–Gordon field is very similar to the real Klein–Gordonfield, except that now the particles created and annihilated by the field carrya charge. Our starting point is the Lagrangian density

L = : !†,µ(x)!,µ(x) ! m2!†(x)!(x) : . (13.2.1)

In line with the remark following Eq. (12.2.24), !(x) and !†(x) are treated asindependent fields. Hence, we have, for example, !L

!"†,µ(x)

= !,µ(x), and, fromthe Euler–Lagrange equations (12.2.15), the equations of motion

("µ"µ + m2)!(x) = 0 and ("µ"µ + m2)!†(x) = 0 . (13.2.2)

The conjugate fields of !(x) and !†(x) are, according to (12.2.16),

#(x) = !†(x) and #†(x) = !(x) . (13.2.3)

Since the complex Klein–Gordon field also behaves as a scalar under Lorentztransformations, it has spin = 0. Due to the gauge invariance of L, this fieldpossesses an additional conserved quantity, namely the charge Q. The equaltime commutators of the fields and their adjoints are, according to canonicalquantization (12.3.1),

!!(x, t), !†(x!, t)

"= i $(x ! x!)

!!†(x, t), !(x!, t)

"= i $(x ! x!)

(13.2.4)

and!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"

=!!(x, t), !(x!, t)

"=

!!(x, t), !†(x!, t)

"= 0 .

The solutions of the field equations (13.2.2) for the complex Klein–Gordonfield are also of the form e±ikx, so that the expansion of the field operatortakes the form

!(x) = !+(x) + !"(x) =#

k

1(2V %k)1/2

$ake"ikx + b†keikx

%, (13.2.5a)

where, in contrast to the real Klein–Gordon field, the amplitudes b†k and ak

are now independent of one another. From (13.2.5a) we havewhere, in contrast to the real Klein–Gordon field, the amplitudes b†k and ak are now independent of one another.

We have286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

inverse Fourier

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

286 13. Free Fields

!†(x) = !†+(x) + !†!(x) =!

k

1(2V "k)1/2

"bk e!ikx + a†

keikx#

.

(13.2.5b)

In (13.2.5a,b), the operators !(x) and !†(x) are split into their positive(e!ikx) and negative (eikx) frequency components. Taking the inverse of theFourier series (13.2.5a,b) and using (13.2.4), one finds the commutation re-lations

$ak, a†

k!

%=

$bk , b†k!

%= #kk!

$ak, ak!

%=

$bk, bk!

%=

$ak, bk!

%=

$ak, b†k!

%= 0 .

(13.2.6)

One now has two occupation-number operators, for particles a and for par-ticles b

nak = a†kak and nbk = b†kbk . (13.2.7)

The operators a†k , ak create and annihilate particles of type a, whilst b†k , bk

create and annihilate particles of type b , in each case the wave vector beingk. The vacuum state |0! is defined by

ak |0! = bk |0! = 0 for all k , (13.2.8a)

or, equivalently,

!+(x) |0! = !†+(x) |0! = 0 for all x . (13.2.8b)

One obtains for the four-momentum

Pµ =!

k

kµ (nak + nbk) , (13.2.9)

whose zero component, having k0 = "k, represents the Hamiltonian. Onaccount of the invariance of the Lagrangian density under gauge transforma-tions of the first kind, the charge

Q = "iq

&d3x : !†(x)!(x) " !(x)!†(x) : (13.2.10)

is conserved. The corresponding four-current-density is of the form

jµ(x) = "iq

':

$!†

$xµ! " $!

$xµ!† :

((13.2.11)

and satisfies the continuity equation

jµ,µ = 0 . (13.2.12)

Substituting the Fourier expansions into Q, one obtains

13.3 Quantization of the Dirac Field 287

Substituting the expansions (13.2.5a,b) into Q, one obtains

Q = q!

k

(nak ! nbk) . (13.2.13)

The charge operator commutes with the Hamiltonian. The a particles havecharge q, and the b particles charge !q. Except for the sign of their charge,these particles have identical properties. The interchange a " b changes onlythe sign of Q. In relativistic quantum field theory, every charged particle isautomatically accompanied by an antiparticle carrying opposite charge. Thisis a general result in field theory and also applies to particles with other spinvalues. It is also confirmed by experiment.

An example of a particle–antiparticle pair are the charged ! mesons !+

and !! which have electric charges +e0 and !e0 . However, the charge neednot necessarily be an electrical charge: The electrically neutral K0 mesonhas an antiparticle K0, which is also electrically neutral. These two particlescarry opposite hypercharge: Y = 1 for the K0 and Y = !1 for the K0, andare described by a complex Klein–Gordon field. The hypercharge1 is a charge-like intrinsic degree of freedom, which is related to other intrinsic quantumnumbers, namely the electrical charge Q, the isospin Iz , the strangeness S,and the baryon number N , by

Y = 2(Q ! Iz)

and

S = Y ! N .

The hypercharge is conserved for the strong, but not for the weak interactions.However, since the latter is weaker by a factor of about 10!12, the hyperchargeis very nearly conserved. The electrical charge is always conserved perfectly!The physical significance of the charge of a free field will become apparentwhen we consider the interaction with other fields. The sign and magnitudeof the charge will then play a role.

13.3 Quantization of the Dirac Field

13.3.1 Field Equations

The quantized Klein–Gordon equation provides a description of mesons and,simultaneously, di!culties in its interpretation as a quantum-mechanicalwave equation were overcome. Similarly, we shall consider the Dirac equa-tion (5.3.20) as a classical field equation to be quantized:

1 See, e.g., E. Segre, Nuclei and Particles, 2nd ed., Benjamin/Cummins, London(1977), O. Nachtmann, Elementary Particle Physics, Springer, Heidelberg (1990)

13.3 Quantization of the Dirac Field 287

Substituting the expansions (13.2.5a,b) into Q, one obtains

Q = q!

k

(nak ! nbk) . (13.2.13)

The charge operator commutes with the Hamiltonian. The a particles havecharge q, and the b particles charge !q. Except for the sign of their charge,these particles have identical properties. The interchange a " b changes onlythe sign of Q. In relativistic quantum field theory, every charged particle isautomatically accompanied by an antiparticle carrying opposite charge. Thisis a general result in field theory and also applies to particles with other spinvalues. It is also confirmed by experiment.

An example of a particle–antiparticle pair are the charged ! mesons !+

and !! which have electric charges +e0 and !e0 . However, the charge neednot necessarily be an electrical charge: The electrically neutral K0 mesonhas an antiparticle K0, which is also electrically neutral. These two particlescarry opposite hypercharge: Y = 1 for the K0 and Y = !1 for the K0, andare described by a complex Klein–Gordon field. The hypercharge1 is a charge-like intrinsic degree of freedom, which is related to other intrinsic quantumnumbers, namely the electrical charge Q, the isospin Iz , the strangeness S,and the baryon number N , by

Y = 2(Q ! Iz)

and

S = Y ! N .

The hypercharge is conserved for the strong, but not for the weak interactions.However, since the latter is weaker by a factor of about 10!12, the hyperchargeis very nearly conserved. The electrical charge is always conserved perfectly!The physical significance of the charge of a free field will become apparentwhen we consider the interaction with other fields. The sign and magnitudeof the charge will then play a role.

13.3 Quantization of the Dirac Field

13.3.1 Field Equations

The quantized Klein–Gordon equation provides a description of mesons and,simultaneously, di!culties in its interpretation as a quantum-mechanicalwave equation were overcome. Similarly, we shall consider the Dirac equa-tion (5.3.20) as a classical field equation to be quantized:

1 See, e.g., E. Segre, Nuclei and Particles, 2nd ed., Benjamin/Cummins, London(1977), O. Nachtmann, Elementary Particle Physics, Springer, Heidelberg (1990)