Homeworks (Problems)
Problem
Prove the following identities (A and B are constant vectors):
∇·(B(A·r)) = A·B ,
∇×(B(A·r)) = A×B ,
∇·(r(A·r)) = 4A·r ,
∇×(r(A·r)) = A×r ,
!
!
r·A
r·A
= ∇×
,
∇
r3
r3
Problem
Prove the following identities
Z
Γ=∂Σ2
Z
dl φ =
Σ2
ds×∇φ ,
Z
Z
r
φ
1
3
ds ,
d r ∇φ = d r φ 3 +
r
r
r
Ω
∂Ω
Ω
Z
Z
Z
1
r·A
A
d3 r ∇·A = d3 r 3 +
ds · ,
r
r
r
Ω
∂Ω
ZΩ
Z
Z
Ω
Z
Ω
Z
Σ
3
d3 r ∇φ =
∂Ω
d3 r ∇×V =
ds = −
1
2
Z
∂Σ
ds φ ,
Z
∂Ω
ds ×V ,
dl ×r
Using one of these identities show that
Z
Σ=∂Ω
ds = 0 .
Problem
√
Treating ln r (r = x2 + y 2 as a distribution find in two dimensions ∇2 ln r.
1
Problem
µν
Starting from the canonical energy momentum tensor Tcan
of the free electromagnetic
µν
field construct the Belinfante symmetric tensor Tsymm (for the procedure, see the
µν
notes attached below). Check by direct calculation that ∂µ Tsymm
= 0.
Problem
µρλ
Following the procedure outlined in my notes construct the canonical tensor Mcan
(Noether current) associated with the invariance of the free electromagnetic field
action I with respect to proper Lorentz transformations
xµ → x′µ = Λµν xν ,
Aν (x) → A′ν (x′ ) = Λνκ Aκ (x) ,
where
Λµν
ρλ
− 2i ωρλ Jvec
= e
µ
ν
,
ρλ µ
(Jvec
) ν = i(g ρµ g λν − g ρν g λµ ) .
µρλ
Check directly that ∂µ Mcan
= 0.
Problem
µρλ
Check that the tensor Mcan
differs by a total four-divergence from the tensor M µρλ =
µλ
µρ
µρ
xρ Tsymm
− xλ Tsymm
. Observe that ∂µ M µρλ = 0 follows now from ∂µ Tsymm
= 0.
Problem
The energy momentum tensor of a field φ can be also obtained by varying its generally covariant action with respect to the metric:
T µν (x) =
δ
I[g, φ] .
δgµν
Obtain in this way the energy momentum tensor of the electromagnetic field whose
generally covariant action reads
1Z 4 q
d x detg gµν gλρ F µλ F νρ .
I=−
4
The necessary trick for varing the determinant of the metric can be found in my
notes on vector analysis.
2
Problem
Check that the action I of the free electromagnetic field is scale invariant, that is,
invarian with respect to transformations
xµ → x′µ = eλ xµ ,
Aν (x) → A′ν (x′ ) = e−λ Aν (x) ,
(λ is an arbitrary real number) and using the Noether theorem derive the associated
µ
µ
symmetry current Jscale
(x). Check by direct calculation that ∂µ Jscale
(x) = 0.
Problem
Show that the freedom in the construction of the conserved Noether currents allows
µ
µ
µρ
to modify the current Jscale
obtained above so that it takes the form J˜scale
= xρ Tsymm
,
µ
µρ
µρ
so that ∂µ J˜scale = 0 is equivalent to the tracelessness of Tsymm : gρµ Tsymm = 0.
3
Notes to classical electrodynamics
I. Special theory of relativity (ref.: Landau & Lifszyc)
Description of physical phenomena requires refrence frames. Inertial frames are singled out because motions of free particles look simple in them. Relativity principle:
All inertial frames are equally good i.e. laws of physics look formally the same in
all inertial frames. This all was Galileo/Newton.
Einstein’s postulate: All inertial frames are equivalent as previously, but there
are no instantaneous interactions - there exist a maximal velocity. Logically, if there
is a maximal velocity, it must be the same in all pysically equivalent frames (inertial
ones), otherwise it wouldn’t be maximal (could be increased by going over to another
frame moving appropriately). This maximal velocity turns out to be the speed of
light (an experimental fact).
In Newtons mechanics distance between events was relative (depending on the
frame) but time separation of events was absolute (same in all frames). With the
Einstein’s postulate also time separation of events must become relative (frame
dependent). This means that two events (separated spatially) holding at the same
time in the frame O do not hold at the same time in a frame O′ moving with respect
to O.
Constant speed of light c in two frames. In O
c2 (t2 − t1 )2 − (x2 − x1 )2 = 0
Infinitesimally
ds2 ≡ c2 dt2 − dx2 = 0
If the speed of light is truly constant, in O′ we have
ds′2 ≡ c2 dt′2 − dx′2 = 0
with the same c. Thus ds2 = 0 implies ds′2 = 0. So they should be proportional to
each other. A priori
ds′2 = f (t, x, V)ds2
where V is the relative velocity of the frame O′ w.r.t. O. This leads to the conformal
symmetry. In the special theory of relativity f (t, x, V) = a(|V|) (since a is a number,
4
not a vector, it can only depend on |V|. One then considers three frames O1 , O2
and O3 with relative velocities V21 , V32 , and V31 , respectively. Then
ds22 = a(|V21 |)ds21 ,
ds23 = a(|V32 |)ds22 ,
ds23 = a(|V31 |)ds21,
which implies
a(|V31 |) = a(|V32 |) a(|V21|)
This must hold for all possible orientations of the velocities V32 , V21 which can only
hold for a ≡ 1. The quantity ∆s2 ≡ c2 (t2 − t1 )2 − (x2 − x1 )2 is called the interval
between the two events (t2 , x2 ) and (t1 , x1 ).
From the interval the formula for the proper time elapsed in a moving frame seen
from another frame readily follows. dx′2 = 0 (because we talk about a clock at rest
in the moving frame) and, hence,
2
′2
2
2
2
2
c dt ≡ c dτ = c dt
v2
1− 2
c
!
or
∆τ =
Z
t2
t1
s
dt 1 −
v2
c2
It follows that ∆τ ≤ ∆t. This is called the time dilatation.
Lorentz contraction. Consider two events holding in O′ at the same moment and
separated by ∆x′ = l0 . These “events” can correspond to two ends of a rod; the
time variable enters the game because what we call l0 is the distance between the
rod’s ends at the same moment in O′ . Now, if O′ moves with respect to O with
velocity V then the rod’s ends mark two world-lines in O. In this frame we then
have
V
′
′
∆x = γ ∆x + ∆ct
c
V
′
′
∆ct = γ ∆ct + ∆x
c
What we call the lenght l of the rod in O is the difference of the positions of its ends
at the same moment in O. Hence ∆ct = 0 i.e. ∆ct′ = −(V /c)∆x′ and putting this
into the first formula above we get
l ≡ (∆x)∆ct=0 =
s
V2
1 − 2 ∆x′ ≡
c
5
s
1−
V2
l0
c2
It follows that l ≤ l0 .
Lorentz transformations must preserve the interval. Suppose O′ moves in O along
the x axis with V . Then the possible linear transformation of time and coordinates
must be of the form
ct = ct′ chψ + x′ shψ
x = x′ chψ + ct′ shψ
because ch2 ψ −sh2 ψ = 1. To find the meaning of ψ consider the motion of the origin
of O′ in O. Hence x′ = 0 and one gets
x
V
≡
= thψ
ct
c
or
1
chψ = q
1 − V 2 /c2
V /c
,
shψ = q
1 − V 2 /c2
Thus ψ is related to V . Explicitly now the transformation, called Lorentz transformation reads
1
V
ct + x′
c
′
ct = q
1 − V 2 /c2
1
V
x=q
x′ + ct′
c
1 − V 2 /c2
The inverse transformations read
V
ct − x
c
1
′
ct = q
1 − V 2 /c2
V
1
x − ct
x′ = q
c
1 − V 2 /c2
For |V | ≪ c obviously t′ ≈ t and x′ ≈ x − V t.
More generally, it is convenient to introduce the four-vectors
xµ = (ct, x, y, z),
and
xµ ≡ (ct, −x, −y, −z) = gµν xν
6
and to write the interval in the form
∆s2 = gµν ∆xµ ∆xν
and Lorentz transformations as
x′µ = Λµν xν
The matrices Λ are the matrices satisfying the following relation
gµν Λµρ Λνκ = gρκ
or, in the matrix form,
ΛT gΛ = g
They form the vector representation of the pseudo-orthogonal group O(1, 3) called
the Lorentz group. The rotation group O(3) is obviously its subgroup. You can
expand on this point a little bit, for example counting the number of parameters
and introducing the name boost.
Transformations of velocities. If a body moves w.r.t O′ with the velocity v′ =
(v ′x , v ′y , v ′z ), and in O the frame O′ moves in the x direction with velocity V then
what is the velocity of the body in O? Using the Lorentz transformation of the
differentials dy = dy ′, dz = dz ′ and
dividing one gets
1
dx = q
(dx′ + V dt′ )
1 − V 2 /c2
1
V ′
′
dct = q
dct + x
c
1 − V 2 /c2
dx
v x′ + V
≡ vx =
,
dt
1 + V v x′/c2
dy
≡ vy =
dt
s
1−
v x′
V2
c2 1 + V v x′ /c2
Its important that if |v′| < c then |v < c, consistently with the Einstein’s postulate.
One should also stress the precise definition of this law of the composition of velocities: it says what is the velocity in O of a body moving with v′ with respect to O′ .
7
< c and another particle coming from
If I see a particle coming from the left with v1 ∼
< c (both along the x axis) then their relative velocity
the right with |v2 | = −v2 ∼
< 2c. There is no contradiction with SR here. If
as defined in my frame is v1 − v2 ∼
I want to know the time after which these particles will collide, I must divide the
< 2c.
distance between them (measured in my frame!) by |v1 − v2 | ∼
One can introduce the four-velocity uµ and the four-acceleration
uµ =
dxµ
,
dτ
aµ =
d2 xµ
dτ 2
Explicitly

c
v


uµ =  q
, q
1 − v 2 /c2
1 − v 2 /c2
Obviously uµ uµ = c2 . Hence, aµ uµ = 0.
Four-momentum pµ = (E/c, px , py , pz ).
mc2
E=q
2
1 − vc2
mv
p= q
,
2
1 − vc2
It follows that E 2 = c2 p2 + m2 c4 and v = c2 p/E. Transformation laws
1
E V x
E′
=q
− p
c
c
1 − V 2 /c2 c
1
V E
x′
x
q
p =
p −
c c
1 − V 2 /c2
8
II Symmetries and the Noether Theorem
Dynamics of a system of classical fields φi is determined by the action I[φ]
which is a functional over all differentiable space-time field configurations. Actions
considered in connection with quantum field theory are given in terms of Lagrangians
L obtained as space integrals of local Lagrangian densities
I[φ] =
Z
t1
t0
dt
Z
V
d3 x L(φ, ∂µ φ) ≡
Z
t1
t0
dt L .
(1)
The Lagrangian density L is a function of fields φi , i = 1, . . . , N and their spacetime derivatives. In the following we consider Lagrangian densities depending on first
order derivatives only. The volume V may be finite or infinite; in the latter case one
assumes that all the fields φi vanish sufficiently fast at spatial infinity; in the finite
volume V some spatial boundary conditions must be specified. The action I[φ] must
be a real and - for relativistic field theories - Poincaré invariant quantity. In general
the fields φi transform nontrivially (as some, in general reducible, representation of
the SO(1, 3) group or, in the case of fermionic fields to be discussed in section ??,
of its universal covering SL(2, C)) under changes of the inertial frame and can also
transform nontrivially under some internal symmetries.
Equations of motion of a system of fields are obtained by requiring that for the
true field configurations the action (1) is stationary (i.e. it does not change to first
order in δφi (x)) with respect to substitutions φi (x) → φi (x) + δφi (x), where the
variations δφi (x) are assumed to vanish for t = t1 and t = t2 as well as for |x| → ∞
(if the volume V in (1) is finite, δφi (x) are assumed to vanish at the boundaries).
Concisely this requirement is written as δI = 0. For Lagrangian densities which
depend only on fields and their first derivatives the variation δI of the action due
to any variations δφi (x) of fields (not necessarily subject to some specific boundary
conditions) is1
#
"
Z
∂L
∂L
4
δφi +
δ(∂ν φi )
(2)
δI = d x
∂φi
∂(∂ν φi )
(summation over the index i is understood). In the following we will consider variations φi (x) → φi (x) + δφi (x) for which δ(∂ν φi ) = ∂ν (δφi ). The variation δI can be
then rewritten as
δI =
Z
4
dx
("
#
"
∂L
∂L
∂L
δφi + ∂ν
− ∂ν
δφi
∂φi
∂(∂ν φi )
∂(∂ν φi )
1
#)
.
(3)
Generalization to Lagrangian densities depending on higher field derivatives is straightforward
but r equires assuming appropriate boundary conditions also for derivatives of field variations.
9
Due to the boundary conditions imposed on the variations δφi considered in determination of the true field configurations, the second, surface term in (3) vanishes
and the condition δI = 0 leads to the Euler-Lagrange equations of motions:
∂ν
∂L(φ, ∂φ)
∂L(φ, ∂φ)
−
= 0.
∂(∂ν φi )
∂φi
(4)
It is important to remember that Lagrangian densities which differ from each other
by a total derivative of an arbitrary function of fields2
L′ = L + ∂µ X µ (φ) ,
(5)
give the same classical equations of motion (due to vanishing of the field variations
on the boundary).
Formulation of the field dynamics in terms of the action I allows to easily identify
symmetries of the field equations of motion and, via the Noether theorem, of the
corresponding conserved quantities. We begin with the case of symmetries which
do not affect space-time coordinates (these can be ordinary global symmetries or, in
the case of theories involving fermionic fields, rigid supersymmetries). We consider
first a change of the field variables
φi (x) → φ′i (x) = φi (x) + δ0 φi (x) .
(6)
In the new field variables φ′i (x) the dynamics is described by a new Lagrangian
density L′ depending on the fields φ′i , which can be chosen so that3
L′ (φ′ , ∂µ φ′ ) = L(φ, ∂µ φ) .
(7)
2
For Lagrangian densities depending only on fields and their first field derivatives X µ can depend
also on field derivatives but only in such a way that its variation vanishes for vanishing variations
δφi of fields without imposing any conditions on variations of field derivatives.
3
This is just as in classical mechanics in which one is allowed to use any set of dynamical
variables, qi (t) or qi′ (t) = qi′ (q(t), t), to describe a given system. The equations of motion in the
new variables follow then from the new Lagrangian L′
d ∂L′ (q ′ , q̇ ′ , t)
∂L′ (q ′ , q̇ ′ , t)
=
.
dt
∂ q̇i′
∂qi′
Because Lagrangian has physically a well defined interpretation (e.g. L = T − V in nonrelativistic mechanics), the new Lagrangian L′ (q ′ , q̇ ′ , t) is obtained just by inverting the relations
qi′ = qi′ (q(t), t) and inserting qi (t) = qi (q ′ (t), t) into the original Lagrangian: L′ (q ′ (q), q̇ ′ (q), t) =
L(q(q ′ (t), t), q̇(q ′ (t), t), t).
10
In this case
I′ − I ≡
Z
d4 x L′ (φ′ , ∂µ φ′ ) −
Z
d4 x L(φ, ∂µ φ) ≡ 0 ,
(8)
i.e. I ′ = I. Consequently, if I is stationary for a configuration φi (x) of the original
fields, that is, if φi (x) satisfy the equations (4), then I ′ is stationary for φ′i (x) related
to φi (x), by (6), i.e. the configuration φ′i (x) of the primed fields satisfy the equations
∂ν
∂L′ (φ′ , ∂φ′ )
∂L′ (φ′ , ∂φ′ )
−
= 0,
∂(∂ν φ′i )
∂φ′i
(9)
which are in general of different form than the equations (4) because L′ (· , ·) is in
general a different function of its arguments than L(· , ·).
The choice of L′ satisfying the condition (7) is not the only possibility: any L′
such that
L′ (φ′ , ∂µ φ′ ) = L(φ, ∂µ φ) + ∂µ X µ (φ) ,
(10)
is equally good. In this case, instead of (8), one has
I′ = I +
Z
d4 x ∂µ X µ (φ) ,
(11)
but still, if I is stationary for a configuration φi (x), then I ′ is such for φ′i (x) related
R
to φi (x) by (6) because δ d4 x ∂µ X µ (φ) = 0 for variations δφi vanishing for t1 and
t2 and for |x| → ∞.
It is important to realize, that we are not yet speaking about a symmetry of
the field equations of motion, but rather about the well known fact that in the Lagrangian formalism a given system can be described by arbitrarily chosen dynamical
variables. A change of variables4 φi → φ′i (φi ) is a symmetry of the equations of motion if L′ (· , ·) leading to (10) is such that
L′ (· , ·) = L(· , ·) + ∂ µ Xµ′ (·) ,
(12)
or, using the freedom to redefine L′ by subtracting from it ∂ µ Xµ′ , if for L′ (· , ·)
leading to (10) one can just take L(· , ·), because then the equations of motion (4)
4
Mathematically speaking,the fields φi (x), i = 1, . . . , N define mappings from the space-time
into some N -dimensional “target” space T (N ) , called by physicists the internal space. Therefore φi
should be viewed as coordinates of a point in T (N ) onto which the space-time point characterized
by the coordinates xµ is mapped. A change of variables φi → φ′i can be due to changing the
coordinate system of T (N ) , in which case we have to do with a passive transformation, or due to
considering a transformed system (active transformation).
11
satisfied by the fields configuration φi (x) for which I is stationary have the same
form as the equations of motion (9) satisfied by the configuration φ′i (x) for which
I ′ is stationary. In other words, φi (x) and φ′i (x) are both solutions of the same
Euler-Lagrange equations. Thus, the condition that the change φi → φ′i (φi ) is a
symmetry reads
L(φ′, ∂φ′ ) = L(φ, ∂φ) + ∂µ X µ (φ) .
(13)
Considering now an infinitesimal symmetry transformation (in the sense specified
above) φi → φi + δ0 φi , we write
L(φ′ , ∂φ′ ) = L(φ + δ0 φ, ∂φ + δ0 ∂φ) = L(φ, ∂φ) + δL(φ, ∂φ) ,
(14)
with δL of first order in δ0 φi . Using then (3) as well as the condition (13) combined
with (11) we can write the equality5
0≡
=
Z
Z
d4 x [L(φ′ , ∂φ′ ) − L(φ, ∂φ) − ∂µ X µ (φ)]
d4 x
("
#
"
(15)
#
)
∂L
∂L
∂L
δ0 φi + ∂µ
− ∂µ
δ0 φi − δX µ (φ) + . . . .
∂φi
∂(∂µ φi )
∂(∂µ φi )
The first bracket in the above formula vanishes for the fields φi satisfying the equations of motion (4). Thus, for such a fields configuration φi (x) the quantity
jµ =
∂L
δ0 φi − δXµ (φ) ,
∂(∂ µ φi )
(16)
is conserved, i.e. satisfies the equation
∂ µ jµ = 0 ,
(17)
because for a symmetry transformation (15) is identically zero, and no special requirements on the values at t = t1 and t = t2 of the field changes δ0 φi (x) due to the
symmetry transformations are imposed (of course, δ0 φi (x) must vanish at spatial
infinity or, in the finite volume, should not change the boundary conditions satisfied
by the fields).6 If the infinitesimal changes δ0 φi are parametrized by n independent
5
This reasoning can easily be generalized to Lagrangian densities depending on higher field
derivatives.
6
For this reason the second term on the right hand side of (11) is nonvanishing - it receives
contributions from the t = t1 and t = t2 hypersufaces. Notice that the situation is quite different
in the case of theories formulated in the Euclidean space with coordinates x̄µ : because in this
case fields and, therefore, also the symmetry changes δ0 φi are bound to vanish for |x̄| → ∞ in all
directions, from (15) one cannot infer the existence of conserved currents and, hence, there are no
conserved charges (20).
12
infinitesimal constant parameters θa (in theories with fields taking values in Grassmann algebras the parameters θa can be anticommuting Grassmann variables),
δ0 φi = θa δ0a φi ,
(18)
there are n conserved Noether currents
jµa =
∂L
δ a φi − Xµa (φ) ,
∂(∂ µ φi ) 0
a = 1, . . . , n,
(19)
(for (18) δX µ must also be of first order in the transformation parameters, i.e.
δXµ (φ) = θa Xµa (φ)). The Noether charges Qa given by
a
Q =
Z
d3 x j0a (t, x) ,
a = 1, . . . , n,
(20)
are then time independent (conserved) quantities provided the fields fall off sufficiently rapidly at spatial infinity (or, in a finite volume, satisfy the appropriate
boundary conditions).
In most cases the transformations (18) are linear7 in the fields φi
θa δ0a φi = −i θa Tija φj ,
(21)
(summations over a and j are understood) with T a being a set of Hermitian matrices
- a matrix representation of the generators of a symmetry group G - forming a basis
of a representation of the dimension n Lie algebra of G. The matrix generators T a
satisfy the commutation rule
h
i
T a , T b = iT c fc ab ,
(22)
with some structure constants fc ab and will be assumed to be normalized by the
condition tr(T a T b ) = 21 δ ab . The transformations (6) are then infinitesimal forms of
finite field transformations. For such internal symmetries8 Xµa ≡ 0.
7
However, symmetries realized on fields nonlinearly also play an important role, mainly in socalled effective quantum field theories, e.g. in the chiral Lagrangian description of low energy
strong interactions of mesons.
8
More generally, whenever X µ (φ) cannot be removed by a suitable redefinition of L by a total
four-divergence, we have to do with a space-time symmetry (for example, X µ (φ) cannot be removed
for supersymmetric transformations); otherwise we speak about internal symmetries.
13
In order to discuss space-time transformations (like translations, rotations or
Lorentz boosts) the formalism has to be generalized. An infinitesimal such a transformation xµ → x′µ
x′µ = xµ + δxµ (x) = xµ + θa δ a xµ (x) ,
(23)
should be accompanied by a corresponding transformation of the fields9 φi (x):
φ′i (x′ ) = φi (x) + δφi (x) = φi (x) + θa δ a φi (x) .
(24)
For example, for an infinitesimal transformation xµ → x′µ with
x′µ = xµ + ω µν xν − ǫµ ,
(25)
corresponding to a change of the reference frame, the fields φi (x) transform under
some regular (in general reducible) matrix representation (J µν )ij of the Lorentz
group10
φ′i (x′ ) = φi (x) −
i
ωµν (J µν )ij φj (x) .
2
(26)
One can also consider transformations xµ → x′µ which correspond e.g. to infinitesimal conformal transformations etc.
The system of fields can be described in the new space-time coordinates x′ in
terms of the new field variables φ′ (x′ ) if one uses such a new Lagrangian density L′ ,
that for any two fields configurations φi (x) and φ′i (x′ ) related to each other by (23)
and (24)
I′ − I =
Z
Ω′
d4 x′ L′ (φ′ (x′ ), ∂µ′ φ′ (x′ )) −
Z
Ω
d4 x L(φ(x), ∂µ φ(x)) = 0 ,
(27)
where ∂µ′ denotes the derivative with respect to x′µ and Ω′ is the image of Ω under
the change of variables (23). (27) is a sufficient condition for the new fields φ′ (x′ )
obtained via (24) from the solutions φ(x) of the equations of motion (4) following
from L to be solutions of the equations of motion following from L′ . However as
previously, the same conclusion for φ′i (x′ ) follows if L′ (φ′ (x′ ), ∂µ′ φ′ (x′ )) is chosen such
that
′
I =I+
Z
Ω
d4 x ∂µ X µ (φ) .
9
(28)
This means that the change of the space-time coordinate system entails a related coordinate
change in the internal (target) space.
10
λν µ
given by (58) acting on
The simplest nontrivial is the vector representation with Jvec
κ
vector fields φi ≡ V κ , but higher tensor or spinorial representations can also be considered.
14
for some function X µ (φ).
Again, one speaks about a space-time symmetry, if (by an appropriate choice of
the factor Xµ′ (φ′ )) for the new Lagrangian density L′ (·, ·) leading to (28) one can
take the original L(·, ·):
L′ (φ′ (x′ ), ∂µ′ φ′ (x′ )) = L(φ′ (x′ ), ∂µ′ φ′ (x′ )) ,
(29)
because then the equations of motion satisfied by φ′i (x′ ) (in the space-time coordinates x′µ ) are the same as the equations of motion (in xµ ) satisfied by φi (x). Taking
into account (29), the condition (28) can be written in the form
d4 x′ L(φ′(x′ ), ∂µ′ φ′ (x′ )) = d4 x [L(φ(x), ∂µ φ(x)) + ∂µ X µ (φ(x))] ,
(30)
(d4 x′ 6= d4 x, i.e. det(∂x′ /∂x) 6= 1 for a general transformation (23)).
In a relativistic field theory, the transformations (25) accompanied by (26) should
be symmetries in the above sense. For such transformations det(∂x′ /∂x) = 1 and
the condition (30) takes the form similar to (13), except for different space-time
coordinates on both sides..
Conserved quantities corresponding to symmetry transformations can be found
using the condition (28). To this end it is technically convenient to split the total
change of the field φi (x) as follows
δφi (x) ≡ φ′i (x′ ) − φi (x)
= φ′i (x′ ) − φ′i (x) + φ′i (x) − φi (x) ≡ δxµ ∂µ φi (x) + δ0 φi (x) ,
(to the first order in the transformation parameters θa the fields φ′i have been replaced by φi in the term with δxµ ) and to work with the functional changes of fields
δ0 φi (x) = φ′i (x) − φi (x) for which
δ0 (∂µ φi (x)) = ∂µ (δ0 φi (x)) .
(31)
Thus, to the first order in the parameters θa , we write the transformed fields and
their derivatives as11
φ′i (x′ ) = φi (x) + δ0 φi (x) + δxλ ∂λ φi (x) + O(θ2 ) ,
∂µ′ φ′i (x′ ) = ∂µ φi (x) + ∂µ (δ0 φi (x)) + δxλ ∂λ ∂µ φi (x) + O(θ2 ) ,
∂ν′ ∂µ′ φ′i (x′ )
λ
(32)
2
= ∂ν ∂µ φi (x) + ∂ν ∂µ (δ0 φi (x)) + δx ∂λ ∂ν ∂µ φi (x) + O(θ ) ,
11
From (25) it follows that ∂x′λ /∂xκ = δκλ + ∂(δxλ )/∂xκ , and ∂xλ /∂x′µ = δµλ − ∂(δxλ )/∂xµ .
Thus, ∂µ′ = ∂µ − (∂µ (δxλ ))∂λ and the terms with derivatives of δxλ cancel out in (32).
15
etc.
This allows to represent L(φ′(x′ ), ∂µ′ φ′ (x′ )) in the identity (in which x and x′ are
connected by (23))
0≡
Z
Ω′
4 ′
′
′
d x L(φ (x
), ∂µ′ φ′ (x′ ))
−
Z
Ω
d4 x [L(φ(x), ∂µ φ(x)) + ∂µ X µ (φ)] ,
(33)
following from (28) in the form12
L(φ′(x′ ), ∂µ′ φ′ (x′ )) = L(φ(x), ∂µ φ(x)) + δL ,
(34)
∂L
∂L
δ0 φi +
∂µ δ0 φi + δxµ ∂µ L + O(θ2 ) ,
∂φi
∂(∂µ φi )
(35)
in which
δL =
where
#
"
∂L
∂L
.
+ ∂µ ∂λ φi (x)
δx ∂µ L ≡ δx ∂µ φi (x)
∂φi
∂(∂λ φi )
µ
µ
(36)
The Lagrangian density change (35) can also be rewritten as
"
#
!
∂L
∂L
∂L
δL = δx ∂µ L +
− ∂µ
δ0 φi + ∂µ
δ0 φi .
∂φi
∂(∂µ φi )
∂(∂µ φi )
µ
(37)
We then make in the first integral is (33) the change of the integration variables
d4 x′ → d4 x (upon which Ω′ → Ω) using the Jacobian13
∂x′µ
det
∂xν
!
= det
δµν
∂δxµ (x)
+
∂xν
!
∂δxµ
≈ 1 + tr
∂xν
!
= 1 + ∂µ δxµ ,
(38)
so that finally to first order the integrand of the first term of (33) becomes
d4 x′ L(φ′ (x′ ), ∂µ′ φ′ (x′ )) = d4 x′ [L(φ(x), ∂µ φ(x)) + δL]
= d4 x L(φ(x), ∂µ φ(x)) + d4 x [δL + L ∂µ δxµ ] .
12
(39)
We again assume, that L depends only on fields and their first derivatives. Conserved Noether
charges corresponding to more complicated Lagrangian densities can be derived using similar methods.
13
We use the relation det(1 + A) = exp{tr ln(1 + A)} ≈ 1 + tr(A).
16
For field configurations φi (x) satisfying the equations of motion (4) the identity (33)
then gives:
Z
Ω
"
d4 x L ∂µ δxµ + δxµ ∂µ L + ∂µ
"
!
#
∂L
δ0 φi − ∂µ δX µ (φ) + O(θ2 )
∂(∂µ φi )
#
∂L
= d x ∂µ L δx +
δ0 φi − δX µ (φ) + O(θ2 ) = 0 .
∂(∂µ φi )
Ω
Z
4
µ
(40)
Taking into account the arbitrariness of Ω and reexpressing δ0 φi in terms of δφi ≡
φ′i (x′ ) − φi (x) = δ0 φi + δxµ ∂µ φi we see that conserved is the quantity
"
#
∂L
∂L
J = g ρL−
∂ρ φi δxρ +
δφi − δX µ (φ) .
∂(∂µ φi )
∂(∂µ φi )
µ
µ
(41)
That is, ∂µ J µ (x) = 0 for fields satisfying the equation (4).
We now consider the conserved currents associated with the Poincaré transformations of the form (25). For translations (ωµν = 0 and ǫµ 6= 0) φ′i (x′ ) = φi (x), so
that δφi = 0. The quantity given by (if X µ (φ) = 0, which is usually the case)
µν
Tcan
=
∂L
∂ νφi − g µν L ,
∂(∂µ φi )
(42)
called the canonical energy-momentum tensor is then conserved:
µν
∂µ Tcan
= 0.
(43)
The four constants of motion (i.e. time independent quantities)
Pµ =
Z
0µ
d3 x Tcan
(t, x) ,
(44)
play the role of the total energy P 0 of the system of fields and of its total momentum
νµ
vector P i . It can be shown that if Tcan
is conserved (and only then), P µ given by
(44) transforms as a true four-vector when the reference frame is changed.
In the case of Lagrangian densities which are of the general form
1
L(φi , ∂µ φi ) = ∂µ φi ∂ µ φi − V (φ) ,
(45)
2
where V (φ) is called the field potential, going over to the Hamilton’s formalism, one
finds
1
1
00
(46)
Tcan
= Πi Πi + ∇φi ·∇φi + V (φ) ≡ H ,
2
2
0k
Tcan
= Πi ∂ k φi .
17
In this case it is easy to see that the transformations of the field φi (x) corresponding
to translations in time and space are generated by Poisson brackets:
{P µ , φi (x)}PB = − ∂ µ φi (x) ,
(47)
so that φ′i (x) = φi (x) − {P µ , φi (x)}PB ǫµ .
The canonical energy-momentum tensor (42) is not always symmetric in its inµν
dices µν. However, if instead of Tcan
one uses another tensor
µν
(x) + H µν (x) ,
T µν (x) = Tcan
(48)
where H µν (x) is an arbitrary conserved tensor, ∂µ H µν (x) = 0, such that
Z
d3 x H 0ν (x) = 0 ,
(49)
the associated conserved charges P µ obtained from the new tensor T µν are (for fields
µν
vanishing sufficiently fast at spatial infinity) the same as the ones obtained from Tcan
.
This is so if one takes
H µν (x) = ∂ρ H ρµν (x) ,
(50)
where H ρµν (x) is antisymmetric in its ρµ indices. Using this freedom one can replace
µν
µν
νµ
Tcan
by a symmetric tensor Tsymm
= Tsymm
. For fields transforming nontrivially under
the Lorentz group of particular interest is the Belinfante symmetric tensor obtained
with
H
ρµν
"
∂L
1
(−iJ µν )ij φj
=
2 ∂(∂ρ φi )
#
∂L
∂L
−
(−iJ ρν )ij φj −
(−iJ ρµ )ij φj .
∂(∂µ φi )
∂(∂ν φi )
(51)
It is the Belinfante symmetric energy-momentum tensor which appears as the right
hand side of the Einstein’s equations of General Relativity.
Conserved currents associated with the Lorentz transformations (ǫµ = 0, ω µν 6= 0
in (25)) are derived in a similar way. Using δxµ = ω µν xν and δφi = − 2i ωµν (J µν )ij φj (x)
following from (26) we get (again for X µ = 0)
µνκ
µκ
µν
Mcan
(x) = xν Tcan
− xκ Tcan
+
18
∂L
(−iJ νκ )ij φj .
∂(∂µ φi )
(52)
The conserved (i.e. time independent) charges
J
νκ
=
Z
0νκ
d3 x Mcan
(t, x) ,
(53)
µνκ
are antisymmetric J νκ = J κν . Again, it can be shown that if Mcan
is conserved, J νκ
transform as a true four-dimensional tensor. The spatial components J ij play the
if (53) role of the total angular momentum of the considered system of fields. For
Lagrangian densities of the form (45) it is straightforward to check that through the
Poisson brackets the tensor J νκ generates Lorentz transformations of fields φi (x).
i
h
{J µν , φi (x)}PB = − (xµ ∂ ν − xν ∂ µ )δij + (−iJ µν )ij φj (x) ,
(54)
so that φ′i (x) = φi (x) − 21 ωµν {J µν , φi (x)}PB . It can also be easily shown, that the
tensor (52) differs by a total four-divergence from the tensor
µκ
µν
M µνκ = xν Tsymm
− xκ Tsymm
,
(55)
µν
in which Tsymm
is the Belinfante symmetric energy-momentum tensor obtained from
(51), which gives therefore the same conserved Noether charges J µν (for field configurations satisfying the equations of motion (4) and vanishing at spatial infinity).
19
Here we establish how the parameters ωµν and aµ are related to the familiar
parameters of boosts, rotations and translations. We take here the passive view, i.e.
we consider the same system viewed from two different frames. The infinitesimal
coordinate transformations
x′µ = Λµν xν − aµ ≈ (δ µν + ω µν )xν − aµ ,
(56)
i
λν µ
x′µ ≈ xµ − ωλν Jvec
xκ − aµ ,
κ
2
(57)
take the form
λν
in which Jvec
are the matrix Lorentz group generators in the vectorial representation
λν
Jvec
µ
κ
= i g λµ g νκ − g νµ g λκ .
(58)
λν
Obviously, the matrices Jvec
satisfy the basic commutation rule (??)
h
i
κλ
µν
κν λµ
κµ λν
λν κµ
λµ κν
.
Jvec
, Jvec
= i Jvec
g − Jvec
g − Jvec
g + Jvec
g
Translations
For translations it readily follows from (57) that if xµ are the coordinates of an event
in a frame O, then xµ′ are the coordinates of the same event in the frame O′ which
is shifted with respect to O by a vector a (which in the frame O has components
ai ), and whose clock is late by τ = a0 with respect to the clock of O.
Rotations
If the frame O′ is rotated with respect to O by an angle φ counterclockwise around
the z-axis common for both frames, O and O′ , we have
x′
y′
=
cos φ sin φ
− sin φ cos φ
x
y
≈
x
0 −i
+ iφ
y
i 0
x
+ ...
y
(59)
12
The generator Jvec
given by (58) is the matrix
12
Jvec
µ
κ




=
0

0 −i
i 0
0
and comparing (57) with (59) one finds that


.

ω 12 = −ω12 = ω21 = −ω 21 = φ .
20
(60)
(61)
The matrix which realizes a finite (passive) transformation, obtained by integrating
12
(57) with ω12 6= 0 and Jvec
given by (60) is
z
12
Rz (φ) ≡ e−iω12 Jvec = eiφJvec
1
0
0
0
cos φ sin φ

=
 0 − sin φ cos φ
0
0
0

0
0

.
0
1

(62)
Similarly, if O′ is rotated with respect to O by an angle θ counterclockwise
around the common y-axis, then
x′
z′
=
cos θ
sin θ
− sin θ
cos θ
x
z
≈
x
0 i
+ iθ
z
−i 0
x
+ ...
z
(63)
31
The generator Jvec
given by (58) is the matrix
31
Jvec
µ
κ




=
0
0
0
−i

i

0
.

(64)
Exponentiating (64), one finds
y
31



Ry (θ) ≡ e−iω31 Jvec = eiθJvec = 

1
cos θ
1
sin θ

− sin θ 

cos θ
.

(65)
23
given by (58).
The matrix Rx (α) can be found similarly by exponentiating Jvec
Explicitly the matrix representing the rotation Rẑ (p̂) which produces p̂ = (sin θp cos φp , sin θp sin
out of ẑ = (0, 0, 1) and which enters the standard transformations Lp (??) and (??)
reads
z
y
Rẑ (p̂) = e−iφp Jvec · e−iθp Jvec = Rz−1 (φp ) · Ry−1 (θp )


1
0
1
0
0
0


 0 cos φ
− sin φp 0   0 cos θp

p

=
 0 sin φp
0
cos φp 0   0
0 − sin θp
0
0
0
1
21
0
0
0 sin θp 

.
1
0 
0 cos θp

(66)
Boosts
Let the frame O′ be boosted with respect to O in the x direction with velocity v (as
measured in the frame O). Then
t′ = γ(t − vx) ≈ t − vx + . . .
x′ = γ(x − vt) ≈ x − vt + . . .
(67)
√
01
with γ = 1/ 1 − v 2 . The generator Jvec
is the matrix
01
Jvec
µ
κ
0 i
i 0


=

0
0



.
0
(68)
0
Inserting (68) into (57) and comparing with (67) one finds that
−ω01 = ω 01 ≡ ω x ≈ v .
(69)
Finite (passive) boosts along the x-axis are realized by the matrix
ch ω01
 sh ω

01
x
01
sh ω01
ch ω01

Λ = e−iω01 Jvec = e−iω01 Kvec = 
1
0




,
0
(70)
1
so that the finite boost takes the form
t′ = ch ω01 (t + th ω01 x) ,
x′ = ch ω01 (x + th ω01 t) ,
(71)
from which it follows that
ch ω01 = √
th ω01 = −v ,
03
Similarly, (Jvec
)
µ
κ
1
.
1 − v2
(72)
is the matrix
03
Jvec
µ
κ



0
=
i
0
0
0
0
22
i
0


,

(73)
The active boosts Bz appearing in the standard transformations Lp (??) acting on
the standard four-momentum of a particle of mass M has therefore the explicit form
z
03
Bz (ω03 ) = e−iω03 Jvec = e−iω03 Kvec
1
 0

= ch ω03 
 0
th ω03

0
1
0
0
0 th ω03
0
0 

,
1
0 
0
1

(74)
in which sh ω03 = |p|/M, ch ω z = E(p)/M. In turn Bz entering Lp (??) appropriate
for a massless particle has a similar form with ω03 = ln u = ln(|p|/κ) (so that
ch ω z = (u2 + 1)/2u, sh ω z = (u2 − 1)/2u).
The meaning of the parameters ωµν and aµ is always the same, irrespectively
of the specific representation of the Poincaré group generators. One can check the
formulae given above by considering e.g. the ordinary quantum mechanical scalar
wave function ψ(t, x). If the frame O′ is shifted with respect to O by a vector a
then for a scalar function one should have
ψ ′ (x′ ) = ψ(x) ≡ ψ(x′ + a) ,
(75)
i.e. the shape ψ ′ (·) of the function as determined in O′ should be the same as the
shape of ψ(·) but its maxima, minima etc. should occur for shifted values of its
argument. So, for infinitesimal ai , and renaming x′ → x, one has
ψ ′ (x) = ψ(x) + ai
∂ψ(x)
+ ...
∂xi
(76)
which can be written in the form
ψ ′ (x) = ψ(x) + iai P̂ i ψ(x) + . . .
(77)
with P̂ i the momentum operator
P̂ i = −i
∂
∂
.
≡ +i
i
∂x
∂xi
(78)
Hence, the wave function ψ ′ in the frame O′ is obtained from the wave function ψ
in O by
i
i
i
ψ ′ (x) = e+ia P̂ ψ(x) = e−iai P̂ ψ(x) .
(79)
Similarly, if the clock of O′ is late with respect to the clock of O by τ , that is, if
t′ = t − τ , the time shape of a function ψ ′ must be the same, i.e.
ψ ′ (t′ ) = ψ(t) = ψ(t′ + τ ) .
23
(80)
This implies (dropping the prime) that
ψ ′ (t) = ψ(t + τ ) = ψ(t) − iτ i
∂
ψ(t) + . . . = ψ(t) − iτ Hψ(t) + . . .
∂t
(81)
Thus, with aµ = (τ, ai ) we have
µ
U(1, a) = e−iaµ P .
(82)
If the axes of O′ are rotated counterclockwise with respect to the axes of O by
an angle θ then for a scalar function ψ, using (59) and the same arguments as above,
one must have
ψ ′ (x′ , y ′) = ψ(x, y) = ψ(x′ − θy ′ , y ′ + θx′ ) ,
(83)
or, dropping the primes,
!
∂
∂
ψ(x, y) + . . .
ψ ′ (x, y) = ψ(x, y) + θ x
−y
∂y
∂x
(84)
(where x, y are the contravariant coordinates). This can be written in the form
ψ ′ (x, y) = ψ(x, y) + iθ xP̂ y − y P̂ x ψ(x, y) + . . .
= ψ(x, y) + iθJ z ψ(x, y) + . . .
(85)
Hence,
i
z
U(Rz (θ), 0) = eiθJ = e− 2 ωµν J
24
µν
z
= e−iω12 J .
(86)