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Mathematical Particle Physics |
The most abstract version of the mechanics of systems of particles involve the formulation of the Lagrangian function for the system. Let K denote the kinetic energy of the system and V its potential energy. The Lagrangian function for a discrete system is simply the difference K−V, the difference of its kinetic and potential energy, expressed in terms of the state variables of the system and their time derivatives as well as any parameters affecting the system, such as electrical, magnetic or gravitational field intensities.
The dynamics of the system can be obtained as a set of partial differential equations based upon the Lagrangian function. The differential equations are derived from the principle that the system evolves such as to make the action of the system, ∫Ldt, an extreme.
The calculus of variation provides the conditions on the variation in the state variables over time that results in an extremum of the action ∫Ldt. These conditions are known as the Euler-Lagrange equations.
For example, consider a point particle of mass m in a uniform gravitational field in which the potential energy is given by V=mgz, where z is the height above an arbitrary point and g is a parameter representing the the strength of the gravitational field. The variable z is the only state variable of the system. The Lagrangian L of the system is
Let (dz/dt) be denoted as v. The Euler-Lagrange equation for the system is
This reduces to
Thus the particle moves downward at an accelerating rate that is independent of the mass of the particle.
Let Q be the vector of the state variables of the system, its generalized coordinates and U be the vector of the time rates of change of those variables. An equilibrium point of the system is a value of the state variable, say Q_{0}, such that the corresponding U vector is the zero vector.
The system can be rewritten in terms of state variables which are deviations from the equilibrium point. For vanishingly small deviations the system is linear. The linear system will have a set of oscillations of specific frequencies. These are called the normal modes of the system and its spectrum.
Let L be the Lagrangian density function at a point in space (x, y, z). Then the Lagrangian for the system is
There are corresponding Euler-Lagrange equations which must be fulfilled for the system to achieve an extreme of the action, a minimum, a maximum or a point of inflection.
An electric field can be described by the force on a unit charge at every point in space. This force per unit charge is a 3 dimensional vector. Let the electric field vector be denoted by E. The component of the electric field along the i axis is denoted as E^{i}, where i can be 1, 2 or 3. Likewise the magnetic field is denoted as B and its components as B^{i}. The electromagnetic field can also be specified in terms of a scalar potential function V and a vector potential function A. The relationship between these two representations of an electromagnetic field is
where ∇V is the gradient of the scalar field V and ∇×A is the curl of the vector field A.
A 4 dimensional vector potential A, in which the index runs from 0 to 3, can be defined in which the first (zeroeth) component is the potential function V and the other three are those of A; i.e.,
Therefore −∇V − ∂A/∂t can be represented as something in the nature of a 4 dimensional gradient of A. There is a special notation for this operation which will be covered later.
There are two other variables that are relevant for representing an electromagnetic field, a 3 dimensional current vector J and a scalar charge density ρ. These can be used to create a 4 dimensional vector J=(ρ, J).
The Lagrangian density function for the electromagnetic field is then
The Euler-Lagrange equations for the above Lagrangian density function turn out to be the Maxwell equations for an electromagnetic field.
The generalization and hence more general term for vectors and matrices is tensor. In the notation for tensors there is an important difference between variables with subscripts and variables with superscripts. This also applies to symbols for derivatives.
The symbols for space variable are x^{1}, x^{2} and x^{3}. Time is denoted as the zeroeth variable x^{0}. (Some mathematical physicists denote time as the fourth variable.) Greek letters are used to denote indices that go from 0 to 3, whereas an index that ranges from 1 to 3 is denoted by a Latin letter.
The magnitude of any vector a=(a^{0}, a^{1}, a^{2}, a^{3}) is given by
where the matrix g_{μν} is known as the metric tensor.
The equations are much simplified if a repeated index in a single term is always summed. This is the Einstein convention. Thus
For physical theory about the world the metric tensor is taken to be a diagonal matrix with (+l, −1, −1, −1) on the diagonal.
For the above metric tensor the relationship between a variable with superscripts and one with subscripts is that
The notation for derivatives is especially opaque.
On the other hand,
This means that
The expression
When an index is a Latin letter it means that it goes form 1 to 3. Forexample,
This is the expression that was mentioned previously as being in the nature of a 4 dimensional version of ∇.
Let φ be a scalar field, a quantity defined at all points of space. Consider a system in which its Lagrangian density function depends only on φ and the gradient of φ
For example, φ might be pressure in a container. Then ∂_{μ}φ would give the flow directions.
The action for the system is
where dx^{4}=dtdxdydz.
Now consider a variation on the state variables such that
This means that the variation in the Lagrangian density is
The variation in the action is then
The term δ(∂_{μ}) reduces to ∂_{μ}(δφ). Now consider the second term in the integral
According to Gordon Kane an integration by parts operation will put this integral into the form
That is to say, the above integral can be represented as ∫UdV, which is equal to {UV − ∫VdU}. When UV is evaluated over the boundaries of the region of integration it is zero, again according to Gordon Kane.
The above evaluation of the integral means that
Since δφ is arbitrary, in order for δS to be zero it is necessary that
This is the Euler-Lagrange equation for the scalar field.
Let the Lagrangian for a scalar field φ(x) be
where the parameter m is mass.
The first term is the kinetic energy K and the second term is the negative of the potential energy V; i.e., L = K − V.
The derivative ∂L/∂φ is −m²φ. The derivative ∂L/∂(∂_{μ}) is ½∂^{μ}. Therefore the Euler-Lagrange equation evaluates to
The second term is equivalent to ½[∂_{0}²φ − ∇²φ].
In more conventional notation this is
Therefore the Euler-Lagrange equation for the field is
If m is equal to zero this is the wave equation
If m is not equal to zero it is a form of the Klein-Gordon equation in natural units; i.e., with the speed of light and Planck's constant divided by 2p set equal to unity.
Let E^{i} and B^{i} for i=1,2,3 be the components of the electrical and magnetic field intensities, respectively. Let A^{i} for i=1, 2, 3 be the components of the vector potential for the field.
A two-index antisymmmetric tensor F may be defined for the electromagnetic field as follows.
The tensor ε^{ijk} is such that ε^{ijk}=1 if ijk is an even permuation of 123, is −1 for an odd permutaion and equals zero for any other repeated index.
The tensor F allows the Lagrangian for the electromagnetic field to be written as
Now the analysis will go back to the Lagrangian of the electromagnetic field as
Some notable properties of this Lagrangian are:
(To be continued.)
In addition to the case of a real scalar function shown above, the following are utilized in particle physics.
For this case the complex field function φ satisfies the condition (∂^{μ}∂_{μ} + m²)φ = 0 and the Langrangian density function is
where the asterisk * denotes the complex conjugate.
Let ψ(x) be the two component spinor wave function for a spin-½ fermion and let m be its mass. Then its Lagrangian density function is
where i is the imaginary unit √−1. The γ's are related to the Pauli spin matrices and ψ is the product of γ^{0} and the complex conjugate of ψ. The wave function ψ satisfies the Dirac equation
Consider a system with the Lagrangian
where φ is the field intensity.
The kinetic energy intensity K is given by
with summation over ν.
The potential enegy function is of the form
where μ and λ are parameters of the potential function.
The term λφ^{4} represents an interaction which is said to be of strenth λ. The parameter λ is presumed to be positive.
The potential energy function V has the symmetry
The other way of describing this property is that the potential energy function is invariant under the transformation φ → −φ.
The mass for the system (particle) is determined by the behavior of the potential function near to its minimum value. The minimum value of the potential energy function is also known as the ground state and more obscurely as the vacuum.
If μ is a real number and hence μ² is a positive real number then the potential energy function has the shape shown below and the minimum V is V(0).
For this case the mass of the particle is zero.
On the other hand if μ² is negative then the potential energy function has the following, more interesting shape.
In order for μ² to be negative μ must be an imaginary number. The minimum value of V is obviously not V(0). This case gives rise to a positive mass for the particle represented by the Lagrangian. When the Lagrangian has symmetry but the minimum value of V is not zero the situation is called broken symmetry.
(To be continued.)
It is amazing that anyone would believe that such relatively simple manipulations of Lagrangians could tell us anything about the real world. It is immeasurably more amazing that in fact such analysis does tell us profound things about the real world. However, there is the precedent of the theorems of Emma Noether in which in the most famous one she found that if the Lagrangian of a physical system is invariant under continuous transformations with respect to time and location then energy and momentum are conserved. In the case of the Noether theorems the conservation laws were known, or at least strongly suspected to hold. In the case of the Lagrangian analysis for particle physics previously unknown relationships are being revealed.
Sources:
For material on Yang-Mills fields see Yang-Mills.
For material on Special Unitary Groups and Algebras see SU(2) and SU(3).
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