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Fundamentals of Neutrino Physics and Astrophysics$

Carlo Giunti and Chung W. Kim

Print publication date: 2007

Print ISBN-13: 9780198508717

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198508717.001.0001

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(p.649) Appendix C Lagrangian Theory

(p.649) Appendix C Lagrangian Theory

Source:
Fundamentals of Neutrino Physics and Astrophysics
Publisher:
Oxford University Press

C.1 Variational principle and field equations

Let us consider a set of n real fields ψr(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 Lagrangian113

(C.1) Appendix C Lagrangian Theory
The Lagrangian formalism is more suitable for the study of relativistic field theories than the Hamiltonian formalism, because the Lagrangian is a Lorentz scalar, whereas the Hamiltonian represents the energy of the fields and transforms as the time component of the energy–momentum four-vector.

Let us define the action as

(C.2) Appendix C Lagrangian Theory
where Ω is an arbitrary space-time region. According to the variational principle, the fields must be such that the action is stationary,
(C.3) Appendix C Lagrangian Theory
for infinitesimal variations of the fields that vanish on the hypersurface S surrounding the space-time region Ω. These are variations of the type
(C.4) Appendix C Lagrangian Theory
with
(C.5) Appendix C Lagrangian Theory
The variation of the action under the transformation in eqns (C.4) and (C.5) is
Appendix C Lagrangian Theory
(p.650)
(C.6) Appendix C Lagrangian Theory
By using the Gauss theorem and the constraint in eqn (C.5), one obtains
(C.7) Appendix C Lagrangian Theory
where dS µ(x) is the surface element (see eqn (2.292)). Then, the variational principle in eqn (C.3) leads to
(C.8) Appendix C Lagrangian Theory
Since the variations δvψr(x) are arbitrary and independent for different r, the fields must satisfy the Euler-Lagrange equations
(C.9) Appendix C Lagrangian Theory
It is clear that the scalar character of the Lagrangian is crucial in order to guarantee the Lorentz covariance of the field equations in eqn (C.9).

C.2 Canonical quantization

In quantum field theory the canonical quantization of fields is implemented by imposing on the fields ψr and their canonically conjugated momenta

(C.10) Appendix C Lagrangian Theory
the equal-time relations
(C.11) Appendix C Lagrangian Theory
(C.12) Appendix C Lagrangian Theory
where the plus and minus subscripts denote, respectively, anticommutators for fermion fields and commutators for boson fields.

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 ψr,

(C.13) Appendix C Lagrangian Theory
where ε is infinitesimal and the variation δψr(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:
(C.14) Appendix C Lagrangian Theory
for any I µ(x). From eqn (C.6) with δvψr replaced by εδψr, 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
(C.15) Appendix C Lagrangian Theory
Thus, the transformation in eqn (C.13) is a symmetry if
(C.16) Appendix C Lagrangian Theory
This is the conservation equation
(C.17) Appendix C Lagrangian Theory
for the current
(C.18) Appendix C Lagrangian Theory
The conservation equation (C.17) implies the existence of a charge
(C.19) Appendix C Lagrangian Theory
which is conserved in time:
(C.20) Appendix C Lagrangian Theory

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 d 3 y [ 0 ( t , y ) , ψ r ( t , x ) ] = 0 . In this case, one can derive the commutation relations

(C.21) Appendix C Lagrangian Theory
where the upper and lower signs apply, respectively, to fermion and boson fields. The infinitesimal symmetry transformation in eqn (C.13) is generated by
(C.22) Appendix C Lagrangian Theory

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

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

(C.23) Appendix C Lagrangian Theory
implies the conservation of the energy–momentum four-vector. Since, the variations of the fields ψr(x) and the Lagrangian ℒ (x) are given by
(C.24) Appendix C Lagrangian Theory
in this case, we have
(C.25) Appendix C Lagrangian Theory
Thus, eqn (C.16) leads to
(C.26) Appendix C Lagrangian Theory
Since the variation δx µ is arbitrary, there are four conservation laws:
(C.27) Appendix C Lagrangian Theory
Here 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 Pν = 0), which form the energy-momentum four-vector:
(C.28) Appendix C Lagrangian Theory
If d 3 y [ L 0 ( t , y ) , ψ r ( t , x ) ] = 0 , eqn (C.21) implies that Pν are the generators of space-time translations of quantized fields through the commutator
(C.29) Appendix C Lagrangian Theory
where the upper and lower signs apply, respectively, to fermion and boson fields. Under a finite space-time translation
(C.30) Appendix C Lagrangian Theory
the fields ψr(x) transform according to
(C.31) Appendix C Lagrangian Theory
Choosing δx = x, we obtain the general space-time behavior of the fields with respect to their value in the origin:
(C.32) Appendix C Lagrangian Theory
Obviously, any product of the fields behaves in the same way.

(p.653) C.5 Lorentz transformations

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

(C.33) Appendix C Lagrangian Theory
From eqns (B.65) and (B.67), the variation of the fields is
(C.34) Appendix C Lagrangian Theory
where ψ(x) is the column matrix of the field components in eqn (B.61). Since the variation of the Lagrangian is given by
(C.35) Appendix C Lagrangian Theory
we have, in this case,
(C.36) Appendix C Lagrangian Theory
Taking into account the antisymmetry of ωαβ, eqn (C.16) leads to
(C.37) Appendix C Lagrangian Theory
Since the six independent components of ωαβ are arbitrary, there are six conservation laws,
(C.38) Appendix C Lagrangian Theory
one for each of the six independent components of 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
(C.39) Appendix C Lagrangian Theory
There are six conserved quantities (∂0 Jαβ = 0) given by the six independent components of the antisymmetric angular momentum tensor
(C.40) Appendix C Lagrangian Theory

C.6 Complex fields

If the fields ψr 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 ψr and ψ r ( ψ r in the case of quantized fields). In this case, the expression in eqn (C.18) for the conserved current must be modified to

(C.41) Appendix C Lagrangian Theory
The equations following eqn (C.18) must be modified in a similar way, by adding the contribution obtained from the variation of ψ r . In particular, we have
(C.42) Appendix C Lagrangian Theory
(C.43) Appendix C Lagrangian Theory
(C.44) Appendix C Lagrangian Theory
(C.45) Appendix C Lagrangian Theory
(C.46) Appendix C Lagrangian Theory

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,

(C.47) Appendix C Lagrangian Theory
where θ is an arbitrary parameter. The most well-known conserved charges are the electric charge and the baryon and lepton numbers.

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 ψr(x) and ψ r ( x ) for an infinitesimal δθ are

(C.48) Appendix C Lagrangian Theory
and ∂μ I μ = 0 (because δℒ = 0), the conserved current (∂μ j μ = 0) is
(C.49) Appendix C Lagrangian Theory
and the conserved charge (∂0Q = 0) is
(C.50) Appendix C Lagrangian Theory

(p.655) The global gauge transformation in eqn (C.47) involves a common variation of the phases of all the n fields ψr(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 θa (a = 1, …, N).

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

(C.51) Appendix C Lagrangian Theory
their variation under an infinitesimal transformation parameterized by δθa (a = 1, …, N) is given by
(C.52) Appendix C Lagrangian Theory
with an implicit summation of the index 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
(C.53) Appendix C Lagrangian Theory
where the real numbers f abc are the structure constants of the group115. Since the infinitesimal parameters δθa are arbitrary, Noether’s theorem implies the existence of a conserved current ( μ j a μ = 0 ) for each generator of the group:
(C.54) Appendix C Lagrangian Theory
The corresponding N conserved charges (∂0Qα = 0) are
(C.55) Appendix C Lagrangian Theory
For quantized fields, these conserved charges generate the group transformations in eqn (C.52) through
(C.56) Appendix C Lagrangian Theory
where the upper and lower signs apply, respectively, to fermion and boson fields. Indeed, using the canonical quantization relations in eqns (C.11) and (C.12) one (p.656) can derive the commutation relations in eqn (C.53) for the conserved charges:
(C.57) Appendix C Lagrangian Theory
Thus, the conserved charges form a representation of the generators of the symmetry group.

Notes:

(113) Strictly speaking, ℒ(x) is a Lagrangian density, which is a function of time and space. The corresponding Lagrangian is L(t) = ∫d3 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+, π0, π forms an irreducible representation of the isospin group SU(2)I, whose structure constants are f abc = εabc. The three-dimensional representation of the generators is (L a)jk = - ajk.