The Framework
The basic idea of this chapter is that we get the correct equations of nature, by minimizing something. What could this something be? One thing is for sure: The object mustn’t change under Lorentz transfor-mations, because otherwise we get different laws of nature for differ-ent frames of reference. In mathematical terms this means the object we are searching for must be a scalar, which is an object transforming according to the(0, 0)representation of the Lorentz group. Together with restricting to the simplest possible choice this will be enough to derive the correct equations of nature. Nature likes it simple.
Starting with this idea, we will introduce a framework called the Lagrangian formalism. By minimizing the central object of the the-ory we get the equations of motion that describe the physical system in question. The result of this minimization procedure is called the Euler-Lagrange equations.
The Lagrangian formalism enables us to derive one of the most-important theorems of modern physics: Noether’s Theorem. This theorem reveals the deep connection between symmetries and con-served quantities1. We will use this connection in the next chapter
1A conserved quantity is a quantity that does not change in time. Famous examples are the energy or the momen-tum of a given system. In mathematical terms this means∂tQ=0→Q=const.
to understand how the quantities we measure in experiments can be described by the theory.
4.1 Lagrangian Formalism
The Lagrangian formalism is an incredibly powerful framework2that
2There are of course other frameworks, e.g. the Hamiltonian formalism, which has the Hamiltonian as its central object. The problem with the Hamilto-nian is that it is not Lorentz invariant, because the energy, it represents, is just one component of the covariant energy-impulse vector.
is used in most parts of fundamental physics. It is relatively simple, because the fundamental object, the Lagrangian, is a scalar3. The
for-3A scalar is an object transforming according to the(0, 0)representation of the Lorentz group. This means that it does not change at all under Lorentz transformations.
malism is very useful if one wants to use symmetry considerations.
If we demand the action, the integral over the Lagrangian, to be in-variant under some symmetry transformation, we ensure that the
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dynamics of the system in question respects this symmetry.
4.1.1 Fermat’s Principle
"Whenever any action occurs in nature, the quantity of action em-ployed by this change is the least possible."
- Pierre de Maupertius4
4"Recherche des loix du mouvement"
(1746)
The basic idea of the Lagrangian formalism emerged from Fer-mat’s principle, which states that light always chooses the path q(t) between two points in space that minimizes the time it takes to travel between the points. Mathematically we can write this, if we define the action of a given path q(t)to be
Slight[q(t)] = dt
and our task is to find the specific path q(t)that minimizes the ac-tion5. To find the minimum6of a given function, we take the
deriva-5The action is here simply the integral over the time for a specific path, but in general the action will be a bit more complicated, as we will see in a moment.
6In general, we want to find ex-tremums, which means minimums and maximums. The idea outlined in the next section is capable of finding both. Nevertheless, we will continue to talk about minimums.
tive and set it to zero. Here we want to find the minimum of a func-tional S[q(t)], which means a function S of a function q(t)and we need a new mathematical idea, called variational calculus.
Fig.4.1: Variations of a path with fixed starting- and end-point.
4.1.2 Variational Calculus - the Basic Idea
If we want to develop a new theory capable of finding the minima of functionals, we need to take a step back and think about what characterises a mathematical minimum. The answer of variational calculus is that a minimum is characterised by the neighbourhood of the minimum. For example, let’s find the minimum xminof an ordinary function f(x) = 3x2+x. We start by looking at one specific x = a and take a close look at its neighborhood. Mathematically this means a+ , where denotes an infinitesimal (positive or negative) variation. We put this variation of a into our function f(x):
f(a+ ) =3(a+ )2+ (a+ ) =3(a2+2a + 2) +a+ . If a is a minimum, first order variations in must vanish, because otherwise we can choose to be negative < 0 and then f(a+ ) is smaller than f(a). Therefore, we collect all terms linear in and demand this to be zero
3·2a + =! 0→6a+1=! 0.
So we find the minimum
xmin=a= −1 6 ,
which is of course exactly the same result we get if we take the derivative f(x) = 3x2+x → f(x) = 6x+1 and demand this to be zero. In terms of ordinary functions this is just another way of do-ing the same thdo-ing, but varational calculus is in addition able to find the extrema of functionals. We will see in a moment how this can be done for a general action functional.
The idea of the Lagrangian formalism is that a principle like the Fermat principle for light exists for massive objects, too. Unfortu-nately, massive objects do not simply obey Fermat’s principle, but we can make a more general ansatz
S[q(t)] = Ldt,
whereLis a, in general non-constant, parameter, called the La-grangian. This parameter happens to be constant for light. In general, the Lagrangian is a function of the position q(t)of the object on ques-tion and in addiques-tion, the Lagrangian can depend on the velocity of the object: L = L(q(t),∂t∂q(t)). This will be discussed in more detail
in the next section7. Before we take a closer look at the usage of the 7Our task will be to find the path q(t) with lowest possible action for a given Lagrangian and given initial conditions.
Before we are able to do that, we need to find the correct Lagrangian, describing the physical system in question. Here is where the symmetries we talked about in the last chapters come in handy. Demanding that the Lagrangian is invariant under all transformations of the Lorentz group, will lead us to the correct Lagrangians.
variational calculus idea for a functional like this, we need to talk about two small things.
4.2 Restrictions
As already noted in Chap. 1.1 there are restrictions to our present theories we can’t motivate from first principles. We only know that we must respect these restrictions in order to get a sensible theory.
One important restriction is that we are only allowed to use the lowest possible, non-trivial order derivatives in the Lagrangian. Triv-ial in this context means with no influence of the dynamics of the system, i.e. on the equations of motion. For some theories this will be first order and for others second order. The lowest order of a given theory is determined by the condition that the Lagrangian must be
Lorentz invariant8, because otherwise we would get different equa- 8In fact, the action must be Lorentz invariant, but if the Lagrangian is Lorentz invariant, the action certainly is, too.
tions of motions for different frames of reference. For some theories we can’t get an invariant term with first order derivatives and there-fore second order derivatives are the lowest possible order.
We simply do not know how to work with theories including higher order derivatives and there are deep systematic problems with such theories9. In addition, higher order derivatives in the
La-9These problems are known as Os-trogradski instabilities. The energy in theories with higher order deriva-tives can be arbitrarily negative, which would mean that every state in such theory would always decay into lower energy states. There are no stable states in such theories.
grangian lead to higher order derivatives in the equations of motion and therefore more initial conditions would be required.
It is sometimes claimed that the constraint to first order derivatives is a consequence of our demand to get a local10theory, but this only
10Locality is a consequence of the basis postulates of special relativity, as shown
in Sec. 2.4. rules out an infinite number of derivatives. A non-local interaction is
of the form11
11We will discuss Lagrangians for particle theories, where we search for particle paths and Lagrangians for field theories, where we search for field configurationsΦ(x). This is the topic of the next section.
Φ(x−h)Φ(x), (4.1) that is, two fields interacting with each other at two different points in spacetime with arbitrary distance h. Using the Taylor expansion we can write
which shows that allowing an infinite number of derivatives would result in a non-local interaction theory.
Another restriction is that in order to get a theory describing free (=non-interacting) fields/particles we must stop at the second power order. This means we only include the terms12
12From another perspective, this means again that we only include the lowest possible, non-trivial terms. Terms with Φ0andΦ1are trivial, as we will see later and therefore we use once more only the lowest possible, non-trivial order, now inΦ.
Φ0,Φ1,Φ2
into our considerations. For example, a term of the formΦ2∂μΦ is of third order inΦ and therefore not included in the Lagrangian for our free theory.