# (p.649) Appendix C Lagrangian Theory

# (p.649) Appendix C Lagrangian Theory

# C.1 Variational principle and field equations

Let us consider a set of *n* real fields ψ_{r}(*x*), with r = 1, …, *n* (for example a set of *n* real scalar fields, or the electromagnetic field *A* ^{μ}(*x*) with four components, µ = 0, 1, 2, 3) and a real *Lagrangian*^{113}

Let us define the *action* as

*variational principle*, the fields must be such that the action is stationary,

*S*surrounding the space-time region Ω. These are variations of the type

*S*

_{µ}(

*x*) is the surface element (see eqn (2.292)). Then, the variational principle in eqn (C.3) leads to

_{v}ψ

_{r}(

*x*) are arbitrary and independent for different

*r*, the fields must satisfy the

*Euler-Lagrange equations*

# C.2 Canonical quantization

In quantum field theory the canonical quantization of fields is implemented by imposing on the fields ψ_{r} and their canonically conjugated momenta

# C.3 Noether’s theorem

Noether’s theorem establishes a connection between symmetries under continuous transformations and conservation laws. Let us consider an infinitesimal transformation of the *n* fields ψ_{r},

_{r}(

*x*) is not subject to any constraint, but the fields are required to satisfy the field equations in eqn (C.9). The transformation (p.651) in eqn (C.13) is a symmetry if the field equations in eqn (C.9) remain invariant, i.e. if the action in eqn (C.2) is invariant up to a surface term (the constraint in eqn (C.5) implies that a surface term does not contribute to the field equations). Therefore, the transformation in eqn (C.13) is a symmetry if the Lagrangian remains invariant up to a four-divergence:

*I*

^{µ}(

*x*). From eqn (C.6) with δ

_{v}ψ

_{r}replaced by εδψ

_{r}, by using the field equations in eqn (C.9), one can see that the variation of the Lagrangian under the transformation in eqn (C.13) is

Since all the quantities in the definition of Q in eqn (C.19) are real, in quantum field theory the conserved charge is an Hermitian operator (Q^{†} = Q). Hence, it is a measurable quantity and its constancy in time allows the use of its eigenvalues for the classification of states.

The canonical quantization relations in eqns (C.11) and (C.12) imply that the charge Q is the generator of the symmetry transformation in eqn (C.13) if $\int {\text{d}}^{\text{3}}y\left[{\mathcal{I}}^{0}(t,\overrightarrow{y}),{\psi}_{r}(t,\overrightarrow{x})\right]=0$. In this case, one can derive the commutation relations

# (p.652) C.4 Space-time translations

Using Noether’s theorem, one can show that the symmetry under infinitesimal space-time translations

_{r}(

*x*) and the Lagrangian

*ℒ (x*) are given by

*δx*

^{µ}is arbitrary, there are four conservation laws:

*T*

^{µν}is the

*energy–momentum tensor*, whose columns are the four conserved currents

*T*

^{µ0},

*T*

^{µ1},

*T*

^{µ2},

*T*

^{µ3}. Hence, there are four constant quantities (∂

_{0}P

^{ν}= 0), which form the

*energy-momentum four-vector*:

^{ν}are the generators of space-time translations of quantized fields through the commutator

_{r}(

*x*) transform according to

*δx = x*, we obtain the general space-time behavior of the fields with respect to their value in the origin:

# (p.653) C.5 Lorentz transformations

Under an infinitesimal Lorentz transformation in eqn (B.22) we have

*x*) is the column matrix of the field components in eqn (B.61). Since the variation of the Lagrangian is given by

_{αβ}, eqn (C.16) leads to

_{αβ}are arbitrary, there are six conservation laws,

*J*

^{μαβ}for fixed μ. Indeed,

*J*

^{μαβ}is antisymmetric in the indices α and β. From eqns (B.67) and (C.27),

*J*

^{μαβ}can be written as

_{0}J

^{αβ}= 0) given by the six independent components of the antisymmetric angular momentum tensor

# C.6 Complex fields

If the fields ψ_{r} are complex (non-Hermitian in the case of quantized fields), each field has two degrees of freedom that can be represented by the real and imaginary
(p.654)
parts or ψ_{r} and ${\psi}_{\text{r}}^{\ast}({\psi}_{r}^{\u2020}$ in the case of quantized fields). In this case, the expression in eqn (C.18) for the conserved current must be modified to

# C.7 Global gauge symmetry

According to Noether’s theorem, charge conservation is a consequence of invariance of the Lagrangian (*δℒ* = 0) under phase transformations of complex fields,

The transformations in eqn (C.47) are called *global gauge transformations*, where the adjective *global* indicates that the parameter θ does not depend on space-time. Since the variations of ψ_{r}(*x*) and ${\psi}_{r}^{\ast}(x)$ for an infinitesimal δθ are

_{μ}

*I*

^{μ}= 0 (because

*δℒ*= 0), the conserved current (∂

_{μ}

*j*

^{μ}= 0) is

_{0}Q = 0) is

(p.655)
The global gauge transformation in eqn (C.47) involves a common variation of the phases of all the *n* fields ψ_{r}(*x*). Such a transformation belongs to the abelian group U(1) of continuous phase transformations. Let us now consider a nonabelian Lie group of continuous transformations g of order *N* (for example SU(*N*)^{114}), whose transformations depend on *N* real parameters θ_{a} (a = 1, …, *N*).

In general, the transformations belonging to a nonabelian group g mix the *n* complex fields ψ_{r}(*x*). We assume that a set of fields ψ_{r}(*x*) is a *n*-dimensional irreducible representation of the group g. Writing the *n* fields ψ_{r} in the matrix form

_{a}(

*a*= 1,

*…, N*) is given by

*a*from 1 to

*N*. Here

*L*

_{a}(

*a*= 1,

*…, N*) are the

*n×n*Hermitian matrices which form the

*n*-dimensional representation of the generators of the group and satisfy the commutation relations

*f*

_{abc}are the structure constants of the group

^{115}. Since the infinitesimal parameters δθ

_{a}are arbitrary, Noether’s theorem implies the existence of a conserved current $\left({\partial}_{\mu}{j}_{a}^{\mu}=0\right)$ for each generator of the group:

*N*conserved charges (∂

_{0}Q

_{α}= 0) are

## Notes:

(113)
Strictly speaking, *ℒ(x*) is a Lagrangian density, which is a function of time and space. The corresponding Lagrangian is *L(t)* = ∫d^{3} *xℒ(x*), which is a function of time only. However, in quantum field theory, the Lagrangian density is the main quantity of interest and in this book we omit, for simplicity, the word *density*.

(114)
U(*N*) is the group of unitary transformations of dimension *N* and can be decomposed into the direct product U(*N*) = SU(*N*)×U(1) where SU(*N*) is the group of unitary unimodular transformations of dimension *N*.

(115)
For example, the pion triplet π^{+}, π^{0}, π^{−} forms an irreducible representation of the isospin group SU(2)_{I}, whose structure constants are *f* _{abc} = ε_{abc}. The three-dimensional representation of the generators is (*L* _{a})_{jk} = -*iε* _{ajk}.