≡ ˜pt
−mq= ˜pt−mq. (4.25)
We can see that this quantity depends on the starting time and by choosing the starting time appropriately we can make it zero. Be-cause this quantity is conserved, this conservation law tells us that zero stays zero for all times.
To summarize for particle theories we have the following connec-tions:
• Translational invariance in space⇒conservation of momentum
• Boost invariance30⇒conservation of ˜pt−mq 30Another name for a boost is transla-tion in momentum space, because the transformation q→q+vt, changes the momentum to m ˙q→m(˙q+v).
• Rotational invariance⇒conservation of angular momentum
• Translational invariance in time⇒conservation of energy
Noether’s theorem shows us why those notions31are used in every 31Except for the conserved quantity following from boost invariance.
physical theory of nature in one or another form. As long as we have the usual spacetime symmetries of our physical laws we have momentum, energy and angular momentum as conserved quantities.
It is instructive to have a look at how those notions occur in field theories. For field theories we have two kinds of symmetries. On the one hand, our Lagrangian can be invariant under transformations of spacetime, which means a transformation like a rotation. On the other hand, we can have invariance under transformations of the field itself, which are called internal symmetries.
4.5.2 Noether’s Theorem for Field Theories - Spacetime Symmetries
For fields one has to distinguish between different kinds of changes that can happen under spacetime transformations. Observer S sees the fieldΨ(x)whereas observer S sees the fieldΨ(x). This is the same field, just from another perspective and the two observers do not see the same numerical field components. The two different descriptions are related by the appropriate transformations of the Lorentz group. We introduced in Sec. 3.7.11 the field representation, which we will be using now. The (infinite-dimensional) differential
operator representation changes x → x. This means by using this representation we can compute the field components at a different point in spacetime or in a rotated frame. The finite-dimensional transformation of the Lorentz group changesΨ → Ψ, i.e. mixes the field components32.
32Remember for example, that a Weyl spinor has two- and a vector field has four-components. If we look at
the vector field Aμ = different perspective, i.e. describe it in a rotated coordinate system it can look
like Aμ = Aμdescribe the same field in coordinate systems that are rotated by 90◦around the z-axis relative to each other.
A complete transformation, for a field that depends on spacetime, needs to consider both parts. We will look at these parts separately, starting with the change x→x. For rotations the conserved quantity that follows is not really conserved, because we neglected the sec-ond part of the transformation. Only the sum of the two conserved quantities that follow from x→xandΨ→Ψis conserved.
To make this more concrete consider a general Lagrangian density L((Φ(xμ),∂μΦ(xμ), xμ). Symmetry means we have
L((Φ(xμ),∂μΦ(xμ), xμ) =L((Φ(xμ),∂μΦ(xμ), xμ). (4.26) In general, the total change of a function-of-a-function, when the independent functions are changed and the point at which they are evaluated is also changed, is given by33
33If this is new to you: This is often called the total derivative. The total change is given by the sum of the change rates, also known as derivatives, times the change of the quantity itself.
For example, the total change of a function f(x, y, z)in three dimensional space is given by ∂ f∂xδx+∂ f∂yδy+∂ f∂zδz.
The change rate times the distance it is changed. We consider infinitesimal changes and therefore this can be seen as the first terms in the Taylor expansion, where we can neglect higher order terms.
d f(g(x), h(x), ...) = ∂ f∂gδg+∂ f∂hδh+...+ ∂ f∂xδx. (4.27) Applying this to the Lagrangian yields
δL = ∂L∂ΦδΦ+∂(∂L∂μΦ)δ(∂μΦ) +∂L∂x
μδxμ, (4.28) which we can rewrite using the Euler-Lagrange equations34
34See Eq. 4.10: ∂L∂Φ =∂μ(∂(∂L∂μΦ))
The variationδΦ has now two parts
δΦ= μνSμνΦ(x) − ∂Φ(x)
∂xμ δxμ, (4.30)
with the transformation parameters μν, the transformation operator Sμνin the corresponding finite-dimensional representation and a con-ventional minus sign. Sμνis related to the generators of rotations by Si= 12 ijkSjk and to the generators of boosts by Ki=S0i, analogous to the definition of Mμνin Eq. 3.165. This definition of the quantity Sμν enables us to work with the generators of rotations and boosts at the same time.
The first part is only important for rotations and boosts, because translations do not lead to a mixing of the field components. For boosts the conserved quantity will not be very enlightening, just as in the particle case, so in fact this term will become only relevant for rotational symmetry.
Let’s start with the simplest field transformation: A translation in spacetime, i.e.
xμ→xμ=xμ+δxμ=xμ+aμ. (4.31) From Eq. 4.30 we have for translations, with μν = 0, because field components do not mix under translations
δΦ= −∂x∂Φ
μδxμ
and thus from Eq. 4.29, if we want to investigate the consequences of invariance (δL =0) and we define the energy-momentum tensor
Tμν:= ∂(∂L∂νΦ)∂(Φ)
∂xμ −δνμL . (4.35) Equation 4.34 tells us that Tμνfulfils a continuity equation, because aμ is arbitrary
∂νTμν=0 (4.36)
→∂νTμν=∂0Tμ0−∂iTμi =0 (4.37) for each componentμ. This tells us directly that we have conserved quantities, because for example forμ=0 we get35
35Using∂0 = ∂tand∂iTi0 = ∇T and the famous divergence theorem
Vd3x∇A=
δVd2xA, which enables us to rewrite the integral over some volume V, as an integral over the cor-responding surfaceδV. A very illumi-nating proof of the divergence theorem, there called Gauss’ theorem, can be found athttp://www.feynmanlectures.
caltech.edu/II_03.htmlwhich is chapter 3 of the freely online avail-able Richard P. Feynman, Robert B.
Leighton, and Matthew Sands. The Feynman Lectures on Physics: Volume2.
Addison-Wesley, 1st edition, 2 1977.
ISBN 9780201021172
∂0T00+∂iT0i =0→∂0T00= −∂iT0i
∂tT00= −∇T →
Integrating over some infinite volume V
Divergence theorem;δV cenotes the surface of the volume V
= −
δVd2xT =
Because fields vanish at infinity
0 (4.38)
→∂t
Vd3xT00=0 (4.39)
In the last step we use that if we have an infinite volume V, like a sphere with infinite radius r, and we have to integrate over the surface of this volumeδV, we need to evaluate our fields at r =
∞. We discovered in Sec. 2.3 that we have an upper speed limit for everything in physics. Therefore the field configuration infinitely far away, can’t have any influence on physics at a finite x and we say that fields vanish at infinity.
We conclude: The invariance under translations in spacetime leads us to 4 conserved quantities36, one for each componentμ . Equation
36Because∂0Tμ0aμ =∂0T00a0−∂0Ti0ai = 0, with arbitrary aμwe get a separate continuity equation for each compo-nent.
4.39 tells us these are
E= d3xT00 (4.40)
Pi = d3xTi0, (4.41) where as always i = 1, 2, 3. These quantities are called the total en-ergy E of the system, which is conserved because we have invariance under time-translations x0→x0+a0and the total momentum of the field configuration Pi, which is conserved because we have invariance under spatial-translations xi →xi+ai.