which shows that the canonical energy-momentum tensor is in general not symmetric in the indices µ andν.
Now we make the ansatz
Tµν = Θµν+∂ρωρµν, (3.53)
where ωρµν is an arbitrary tensor field which is antisymmetric in the indicesρ and ν.
Now we try to chose ωρµν such that Tµν is symmetric. Since it differs from the canonical energy-momentum tensor only by a total divergence it yields the same total energy and momentum for the field configuration. The antisymmetry of ωρµν in ρ and ν makes the divergence of Tµν vanish the same time with Θµν.
Inserting this ansatz in (3.53) shows that it is consistent with setting
∂ρ(ωρµν−ωρνµ) =∂ρ ∂L
∂(∂ρφ)σˆµνφ
. (3.54)
The general solution of this equation is given by ωρµν−ωρνµ = ∂L
∂(∂ρφ)ˆσµνφ+∂σησρµν :=ηρµν, (3.55) whereησρµν is an arbitrary tensor field which is antisymmetric inσ and ρas well as inµand ν. It is clear that then ηρµν is antisymmetric inµ and ν.
Now using
ωρµν −ωρνµ =ηρµν, ωρµν +ωµρν = 0 (3.56) we find that with given ηρµν (3.55) is solved uniquely by
ωρµν = 1
2[ηρµν+ηµνρ−ηνρµ]. (3.57) It is simple to show by an algebraic calculation that indeedωfulfils the conditions we derived for it above. So we find the theorem, proven first by Belinfante in 1939, that we can always find a symmetric energy-momentum tensor.
We shall see in the next chapter that by a clever choice of ησρµν which is the only freedom we have to make the energy-momentum tensor symmetric, makes the energy-momentum tensor of the electromagnetic field gauge-invariant the same time.
3.4 Canonical Quantisation
Now we like to solve our problem with the particle interpretation and causality raised by the negative energy states. For this purpose let us consider a free complex scalar field with the Lagrangian
L = (∂µφ)∗(∂µφ)−m2φ∗φ. (3.58) Although there seems to be no solution in terms of a Schr¨odinger-like theory, i.e., to interpret theφ-field as one-particle wave function, we try to build a many-particle theory by quantising the fields.
For this purpose we need another formulation of the classical field theory, namely the Hamil-tonian one, known from point mechanics and the canonical quantisation as an approach to Dirac’s operator formulation of quantum mechanics in terms of the algebra of observables.
This approach has the disadvantage to destroy manifest Lorentz invariance since we have to introduce canonical momentum densities by the definition
Π = ∂L
∂(∂0φ), Π∗ = ∂L
∂(∂0φ∗). (3.59)
Now let us look on the variation of the action S[φ] =
where the nabla symbol ∇ acts only on the space components of x. This is what is meant by saying that the formalism is not manifest covariant since we have now fixed the reference frame by splitting in space and time components.
But now we can proceed in the same way as we would do in the case of point mechanics: We introduce the Hamiltonian density
H = Π∂tφ−L. (3.61)
Now varying with respect to the fields we find
δH =δΠ∂tφ+ Πδ∂tφ+ cc.−δL =δΠ∂tφ−∂L
∂φ δφ−δ(∇φ) ∂L
∂(∇φ) + cc. (3.62) This shows that the natural variables for H are Π, φ, ∇φ and their conjugate complex counterparts. Now we define the Hamiltonian
H= Z
d3~xH. (3.63)
With help of (3.64) we find for the functional derivative of H, where t is to be seen as a parameter:
and using the definition of H together with the equations of motion (3.29) we find the Hamiltonian equations of motion
∂tΠ =−δH
δφ, ∂tφ= δH
δΠ. (3.65)
For an arbitrary observable, which is a functional of φ and Π and may depend explicitly on t, we find Specially we have the following fundamental Poisson brackets:
{φ(t, ~x),Π(t, ~y)}pb=δ3(~x−~y), {Π(t, ~x),Π(t, ~y)}pb= 0, {φ(t, ~x), φ(t, ~y)}pb = 0. (3.67)
3.4 · Canonical Quantisation
It should be kept in mind that the Poisson brackets are only defined for functionals at one instant of time. That means that a Poisson bracket makes only sense if the quantities entering have equal time arguments.
Now we can use the recipe of canonical quantisation. The fields become operators and build together with the unit operator the algebra of observables. As we know from the nonrela-tivistic case, we can quantise fields with help of commutators describing bosonic particles or anti-commutators describing fermionic particles. We shall see soon that in the relativistic case stating some simple physical axioms we can quantise the scalar field only in terms of bosons. The direct translation of the Poisson bracket rules in commutation rules are the fields in the Heisenberg picture, which we shall give capital Greek symbols. Then the canonical commutation relations read:
1
i [Φ(t, ~x),Π(t, ~y)]−=δ(3)(~x−~y), 1
i[Φ(t, ~x),Φ(t, ~y)]−= 1
i [Π(t, ~x),Π(t, ~y)]− = 0. (3.68) The classical Lagrangian for the free case is given by (3.58). Here φand φ∗ have to be seen to represent two independent real field degrees of freedom. Now we like to quantise these free fields. The first step is to define the canonical field momenta:
Π(x) = ∂L
∂[∂tφ(x)] =∂tφ∗(x), Π∗(x) = ∂L
∂[∂tφ∗(x)] =∂tφ(x). (3.69) The canonical non-vanishing commutation relations (3.68) read therefore for these fields
Φ(t, ~x)↔∂tΦ†(t, ~y) =iδ(3)(~x−~y), (3.70) where the symbol↔∂tis defined by
f(t, ~x)↔∂tg(t, ~y) =f(t, ~x)∂tg(t, ~y)−[∂tf(t, ~x)]g(t, ~y). (3.71) The physical field operators have to fulfil the operator equations of motion, i.e.
(2+m2)Φ= 0. (3.72)
In terms of a Fourier decomposition the field can be written as φ(x) =R d3~p
√2ω(~p)(2π)3 [A+(~p) exp(−iω(~p)t+ i~p~x) +A−(~p) exp(+iω(~p)t+ i~p~x)]
with ω(~p) = +p
~
p2+m2, (3.73)
where the normalisation of the fields will be explained later on. Now the second part does not look physical since it seems to describe a particle with a time-like momentum in the negative light-cone, i.e., a particle which moves in the “direction of the past” which is evidently not consistent with causality. We can reinterpret this term with help of a creation operator of another sort of particles with help ofA−(~p) =b†(−~p) and substitution~p→ −~pin the integral:
φ(x) =
Z d3~p p2ω(~p)(2π)3
h
a(~p) exp(−ipx) +b†(~p) exp(ipx)i
p0=ω(~p). (3.74) This is the solution of the problem of negative energy states, the so called Feynman-Stueckel-berg interpretation. This was possible because we have introduced a multi-particle description
with help of the field quantisation, where the annihilation of a negative energy state corre-sponding to the motion of the particle backward in time can be seen as the creation of a state of positive energy moving forward in time with a momentum in the opposite direction. This redefinition is not possible with c-number fields. This is the first place we see explicitely that the relativistic quantum theory is necessarily a multi-particle theory. Now we define
ϕ~q(x) = 1
p2ω(~q)(2π)3exp[−iω(~q)t+ i~q~x]. (3.75) A simple direct calculation shows that
i Z
d3~xϕ∗~q(t, ~x)↔∂tΦ(t, ~x) =a(~p), i Z
d3~xϕ∗~q(t, ~x)↔∂tΦ†(t, ~x) =b(~p). (3.76) With help of the canonical commutation relations (3.70) we find
ha(~p),a†(~q)i
−=δ(3)(~p−~q), h
b(~p),b†(~q)i
−=δ(3)(~p−~q). (3.77) All other commutators between thea- andb-operators vanish. These relations show that the free complex scalar field describes two distinct sorts of particles and that the Hilbert space the annihilation and creation operators operate in is the Fock space ofa- andb-particles. For example the one-particle states are given by
|a~pi=a†(~p)|0i, |b~pi=b†(~p)|0i, (3.78) where|0i is the vacuum of the theory uniquely defined by a(~p)|0i=b(~p)|0i= 0.
We now want to calculate the total energy and momentum operators. We just remember that due to Noether’s theorem for the classical field theory these quantities are given by
Pν = Z
d3~xΘ0ν = Z
d3~x ∂L
∂(∂tφ)∂νφ+ ∂L
∂(∂tφ∗)∂νφ∗−δν0L
, (3.79)
which was given by (3.46) and (3.47) from translation invariance of the action.
Now we take instead of the c-number fields their operator counterparts in a very naive way.
But here arises the problem of operator ordering since the operators multiplied at the same space-time point do not commute. Let us start with three-momentum with just an arbitrary ordering of the operators:
P~ = Z
d3~x[Π(x)∇φ(x) +∇φ†(x)Π†(x)]. (3.80) Using the plain wave representation of the field operators (3.74) we find after some algebra
P~ = 1 2
Z
d3~p ~p[a†(~p)a(~p) +a(~p)a†(~p) +b†(~p)b(~p) +b(~p)b†(~p)]. (3.81) Nowna(~p) =a†(~p)a(~p) is the operator for the number of a-particles per momentum volume with momentum~p(and the same is true for the analogous expression for the b-particles).
Now we fix the ordering of the fields at the same space-time point by the definition that the vacuum expectation value of the momentum should vanish, because if there is no particle the
3.4 · Canonical Quantisation
total momentum should be zero. This cancels the infinite vacuum expectation value of the terms with the creation operator on the right side. This is the first time we have “renor-malised” a physical quantity, namely the vacuum expectation value of the three-momentum.
This definition of the vacuum expectation value can be expressed by the fact that we just change the order of the creation and annihilation operators such that all annihilation operators come to the right and all creation operators to the left. The order of the annihilation or creation operators among themselves is not important since they commute. This is called normal-ordering and is denoted by enclosing the field operators which are intended to be normal-ordered in colons. One should keep in mind that the normal-ordering on field operator products in space-time is by no means a trivial procedure since the fields is the sum of annihilation and creation operators in contrast to the nonrelativistic case. So the final result for the total momentum operator is
P˜ = Z
d3~x~ea:Θ0a(x) :=
Z
d3~p ~p[na(~p) +nb(~p)]. (3.82) Herein the field operator ordering in the canonical energy-momentum tensor is well-defined with help of the normal-ordering colons.
Applying this procedure to the zero component of the total four-momentum, which is the total energy of the system, it is seen that this is given with the canonically quantised Hamiltonian density which has to be normal-ordered again to get rid of the operator ordering problem and to keep the vacuum expectation value of the energy to be zero:
H= Z
d3~x:H :=
Z
d3p ω(~~ p)[na(~p) +nb(~p)]. (3.83) Since the density operators are positive semi-definite the Hamiltonian is bounded from below (it is a positive semi-definite operator) and the vacuum state is the state with the lowest energy.
So we have found a physical sensible interpretation for the free quantised scalar field if we can show that this scalar field obeys the equation of motion of quantum mechanics. For this purpose one has simply to show that the equations of motion for the Heisenberg picture is true for the fields, namely
∂tφ(x) = 1
i [φ(x),H(t)]−, (3.84)
which is simply proved by inserting the plain wave expansion for the field operator (3.74) and (3.83) by use of the commutator relations for the annihilation and creation operators.
Now we have another symmetry for our free field Lagrangian (3.58). This is the invariance underglobal phase transformations of the fields, given by
φ0(x) = exp(−ieα)φ(x), φ0∗(x) = exp(+ieα)φ∗(x), x0 =x withα∈[0,2π]. (3.85) This is the most simple example for an internal symmetry, i.e., a symmetry of the internal structure of the fields which has nothing to do with the symmetries of space and time. It is called a global internal symmetry since the transformation is the same for all space-time points, because the transformation is independent of space and time variables. We shall see that the assumption of alocal internal symmetry, which is called a “gauge symmetry” has far reaching important implications on the structure of the theory. Since the standard model is
a (unfortunately somewhat ugly) gauge theory these theories are the most successful physical meaningful theories in elementary particle physics. Quantum electrodynamics, which we shall study in detail in chapter 5 is the local gauge theory of the here shown global phase invariance and can be seen as the theory, which is the best to experiments agreeing one ever found.
Now we apply Noether’s theorem, which is in the case of global internal symmetries which leave not only the action but also the Lagrangian invariant very simple to treat. At first we take the infinitesimal version of (3.85):
δφ=−ieδαφ, δφ∗ = +ieδαφ, δx= 0⇒δL =δd4x= 0⇒Ωµ= 0. (3.86) The conserved Noether current from this theory is given by (3.41):
jµ(x) =−ieφ∗↔∂
µ
φ. (3.87)
Although we have not coupled the electromagnetic field to our φ-field we say that this will be identified with the electromagnetic current of the scalar particles. In the case of a real field, which meansφ∗=φthis current vanishes identically. This is understandable since in the real field case there is no phase invariance!
The quantised version is again given a unique meaning by normal-ordering:
jµ=−ie:φ†(x)↔∂
µ
φ(x) :. (3.88)
The corresponding conserved quantity is given by integration over the three-dimensional space (see eq. 3.44) and a simple calculation with use of (3.66) yields
Q= Z
d3~xj0(x) =−e Z
d3~p[na(~p)−nb(~p)], (3.89) which shows that thea-particles andb-particles have the same electric chargeewith opposite signs. It should be emphasised that it is alone this phase symmetry which makes this pair of particles special, namely to be particle and the corresponding antiparticle. Without this symmetry there would be no connection between the two independent sort of particles we have called aandb-particles.
The same time this shows that the normalisation of the plain waves is chosen such that we have the simple representation na = a†a (and analogous for the b-particles) for the density operators.
Now we have described the most important physical symmetries and the related quantities.
There are more general mathematical, but also very important topics left out. They are shown from a more general point of view in appendix B, which should be read at the end of this chapter 3. It describes all representations of the Poincar´e group (more precisely we should say its covering group), which have been found to be important in physics so far.