45
2.3. The Maxwell equations: Lorentz covariance and gauge invariance
Either way gives the same answer: it is the conserved charge which determines the particle’s response to the field. By contrast, there are several other
conservation laws that seem to hold in particle physics, such as lepton number
and baryon number, that apparently have no dynamical counterpart (cf the
remarks at the end of section 1.3.6). To determine the baryon number of a
newly produced particle, we have to use B conservation and tot up the total
baryon number on either side of the reaction. As far as we know there is no
baryonic force field.
Thus gauge theories are characterized by a close interrelation between three
conceptual elements: symmetries, conservation laws and dynamics. In fact,
it is now widely believed that the only exact quantum number conservation
laws are those which have an associated gauge theory force field – see comment (i) in section 2.6. Thus one might suspect that baryon number is not
absolutely conserved – as is indeed the case in proposed unified gauge theories of the strong, weak and electromagnetic interactions. In this discussion
we have briefly touched on the connection between two pairs of these three
elements: symmetries ↔ dynamics; and conservation laws ↔ dynamics. The
precise way in which the remaining link is made – between the symmetry
of electromagnetic gauge invariance and the conservation law of charge – is
more technical. We will discuss this connection with the help of simple ideas
from quantum field theory in chapter 7, section 7.4. For the present we continue with our study of the Maxwell equations and, in particular, of the gauge
invariance they exhibit.
2.3 The Maxwell equations: Lorentz covariance and gauge
invariance
In classical electromagnetism, and especially in quantum mechanics, it is convenient to introduce the vector potential A μ (x) in place of the fields E and
B. We write:
B = ∇ × A
(2.10)
∂A
E = −∇V −
(2.11)
∂t
which defines the 3-vector potential A and the scalar potential V . With these
definitions, equations (2.2) and (2.3) are then automatically satisfied.
The origin of gauge invariance in classical electromagnetism lies in the
fact that the potentials A and V are not unique for given physical fields E
and B. The transformations that A and V may undergo while preserving
E and B (and hence the Maxwell equations) unchanged are called gauge
transformations, and the associated invariance of the Maxwell equations is
called gauge invariance.
2.3. The Maxwell equations: Lorentz covariance and gauge invariance
Either way gives the same answer: it is the conserved charge which determines the particle’s response to the field. By contrast, there are several other
conservation laws that seem to hold in particle physics, such as lepton number
and baryon number, that apparently have no dynamical counterpart (cf the
remarks at the end of section 1.3.6). To determine the baryon number of a
newly produced particle, we have to use B conservation and tot up the total
baryon number on either side of the reaction. As far as we know there is no
baryonic force field.
Thus gauge theories are characterized by a close interrelation between three
conceptual elements: symmetries, conservation laws and dynamics. In fact,
it is now widely believed that the only exact quantum number conservation
laws are those which have an associated gauge theory force field – see comment (i) in section 2.6. Thus one might suspect that baryon number is not
absolutely conserved – as is indeed the case in proposed unified gauge theories of the strong, weak and electromagnetic interactions. In this discussion
we have briefly touched on the connection between two pairs of these three
elements: symmetries ↔ dynamics; and conservation laws ↔ dynamics. The
precise way in which the remaining link is made – between the symmetry
of electromagnetic gauge invariance and the conservation law of charge – is
more technical. We will discuss this connection with the help of simple ideas
from quantum field theory in chapter 7, section 7.4. For the present we continue with our study of the Maxwell equations and, in particular, of the gauge
invariance they exhibit.
2.3 The Maxwell equations: Lorentz covariance and gauge
invariance
In classical electromagnetism, and especially in quantum mechanics, it is convenient to introduce the vector potential A μ (x) in place of the fields E and
B. We write:
B = ∇ × A
(2.10)
∂A
E = −∇V −
(2.11)
∂t
which defines the 3-vector potential A and the scalar potential V . With these
definitions, equations (2.2) and (2.3) are then automatically satisfied.
The origin of gauge invariance in classical electromagnetism lies in the
fact that the potentials A and V are not unique for given physical fields E
and B. The transformations that A and V may undergo while preserving
E and B (and hence the Maxwell equations) unchanged are called gauge
transformations, and the associated invariance of the Maxwell equations is
called gauge invariance.