The speed of light c sets the scale for all speeds, so it is common to use natural units, where c = 1 and velocities are fractions.
Natural Units:
\(c = 1\)
\(\hbar = 1\)
Maxwell's Equations:
S.I. Units
\( \overrightarrow{\nabla} \cdot \overrightarrow{E} = \frac{\rho}{\epsilon_{0}} \hspace{2cm} \overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{\partial \overrightarrow{B}}{\partial t} \)
\( \overrightarrow{\nabla} \cdot \overrightarrow{B} = 0 \hspace{2.25cm} \overrightarrow{\nabla} \times \overrightarrow{B} = \mu_{0} \overrightarrow{J} + \frac{1}{c^{2}} \frac{\partial \overrightarrow{E}}{\partial t} \)
Heaviside-Lorentz Units
\( \overrightarrow{\nabla} \cdot \overrightarrow{E} = \rho \hspace{2.4cm} \overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{1}{c} \frac{\partial \overrightarrow{B}}{\partial t} \)
\( \overrightarrow{\nabla} \cdot \overrightarrow{B} = 0 \hspace{2.45cm} \overrightarrow{\nabla} \times \overrightarrow{B} = \frac{1}{c} ( \overrightarrow{J} + \frac{\partial \overrightarrow{E}}{\partial t} ) \)
Natural Units
\( \overrightarrow{\nabla} \cdot \overrightarrow{E} = \rho \hspace{2.4cm} \overrightarrow{\nabla} \times \overrightarrow{E} = - \frac{\partial \overrightarrow{B}}{\partial t} \)
\( \overrightarrow{\nabla} \cdot \overrightarrow{B} = 0 \hspace{2.45cm} \overrightarrow{\nabla} \times \overrightarrow{B} = ( \overrightarrow{J} + \frac{\partial \overrightarrow{E}}{\partial t} ) \)
Other Examples:
Electrostatic Potential: \( V(\overrightarrow{x}) = \frac{q}{4\pi\epsilon_{0}\lvert\overrightarrow{x}\rvert} \Rightarrow V(\overrightarrow{x}) = \frac{q}{4\pi\lvert\overrightarrow{x}\rvert} \)
Fine Structure Constant: \( \alpha = \frac{e^{2}}{4\pi\hbar c} \approx 137 \Rightarrow \frac{e^{2}}{4\pi\hbar} \Rightarrow \frac{e^{2}}{4\pi} \)
Lorentz Transformations
In special relativity there is a universal speed c which is constant in every inertial reference frame S and \(\bar{S}\). This violates the Galilean transformation between relative velocities in Newtonian mechanics, which are replaced by the group of Lorentz transformations.
Correction Factor:
\(\gamma = \frac{1}{\sqrt{1 - \beta^2}}\)
\(\beta = \frac{v}{c}\)
Lorentz boost (x-axis only): (Equation 1)
\(\bar{t} = \gamma (t - \frac{vx}{c^2})\)
\(\bar{x} = \gamma (x - vt)\)
\(\bar{y} = y\)
\(\bar{z} = z\)
Inverse Lorentz boost (x-axis only): (Equation 2)
\(t = \gamma (\bar{t} + \frac{v\bar{x}}{c^2})\)
\(x = \gamma (\bar{x} + v\bar{t})\)
\(y = \bar{y}\)
\(z = \bar{z}\)
Covariance and Invariance
"Covariant" theories are those that transform properly under changes of coordinate systems. Quantum field theory involves relativistic spacetime, which requires its quantities to be Lorentz covariant. Such quantities are various ranks of tensors, which are invariant under coordinate transformations. These are defined in terms of Jacobians and inverse Jacobians as a consequence of the inverse function theorem, whose partial derivatives intuitively mean the rates at which the coordinate indices change with respect to each other.
Scalars: Rank 0 tensors, just a number. (e.g. electric charge, rest mass of electron)
Vectors: Rank 1 tensors, arrows in N dimensions. (e.g. three-vector of Euclidean space, four-vector of Minkowski spacetime)
Tensor: Rank (N,M) tensors, matrices defined to be invariant under coordinate transformations.
Vectors
In classical mechanics the vectors used are called "three-vectors", which means their length in three spatial dimensions is invariant under translations and rotations. The amount that happens to be in each direction changes depending on the orientation of the coordinate axes. In the relativistic mechanics we use four-vectors whose length in the four dimensions of spacetime is invariant under Lorentz transforms, where their different lengths along the space and time axes in other coordinate systems imply time dilation and length contraction.
Vector Notations:
Three Vectors: Arrow italics, Roman indices.
Vector: \(\overrightarrow{x}, \overrightarrow{p}\)
Components: \(\overrightarrow{x}\) = \(x^{i}, i = 1, 2, 3 \)
Four Vectors: Italics, Greek indices.
Vector: x = \((t, \overrightarrow{x})\)
Components: \(x^{\mu}\), \(\mu = 0, 1, 2, 3\)
Time-like Component: \(x^{0}\)
Four-Vector Examples: The form of \( x = (x^{0}, \overrightarrow{x}\)) with \(c = 1\)
Energy-momentum: p = (\(E, \overrightarrow{p}\))
Current Density: j = (\(\rho, \overrightarrow{j}\))
Vector potential: A = (\(V, \overrightarrow{A}\))
Derivative Operator ( \(\partial_{\mu}\) does not imply \( \mu^{th} \) index value )
\( \partial_{\mu} \) is written with an index instead of \( \partial \) only to indicate whether it is operating on lower or upper indices.
\(\partial_{\mu} \equiv \frac{\partial}{\partial x^{\mu}} = (\frac{\partial}{\partial t} , \overrightarrow{\bigtriangledown}) = (\frac{\partial}{\partial t}, \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}) \)
\(\partial_{\mu} \) is lower index, acting on upper index 4-vector \(x^{\mu}\). "naturally lowered"
Integration
The symbol for integrating a four-vector as represented in indices is:
\( \int \mathrm{d}^{4}x = \int \mathrm{d}x^{0}\mathrm{d}x^{1}\mathrm{d}x^{2}\mathrm{d}x^{3} = \int \mathrm{d}t \hspace{0.1cm} \mathrm{d}\overrightarrow{x} \)
Unless otherwise stated, the integral symbol \( \int \) is assumed to be indefinite, ranging from \( -\infty \) to \( +\infty \).
Given a vector \( a = a^{\mu} \) in coordinate system S = {\(x^{\mu}\)}, we have a coordinate transformation {\(x^{\mu}\)} \( \rightarrow \) {\( \bar{x}^{\mu}\)}.
The identification of the coordinate system S and \(\bar{S}\) is by the bar symbol. The index might be altered to imply summation, because the denominator index would be unbound compared to that of the numerator.
Contravariant Vectors: (Upper Level Index)
The vector \(a^{\mu}\) transforms into \(\bar{a}^{\mu}\) with the coordinate changes by the rule:
\( \bar{a}^{\mu} = \underset{\nu}{\Sigma} ( \frac{\partial \bar{x}^{\mu}}{\partial x^{\nu}} ) a^{\nu} \) (Contravariant Transformation)
This represents the vector a in the original coordinate system being proportioned along the new coordinates. Summing up the partial derivatives of each axis against the others in the other reference frame. The symbol \(\nu\) is a dummy index meaning it is merely summation, and could be replaced with any other unused variable.
Covariant Vectors: (Lower Level Index)
Consider the gradient vectors which are producted by taking the partial derivative of a scalar function:
\( \partial_{\mu} \equiv \frac{\partial}{\partial x^{\mu}} \phi \) (Lowered Index Vector)
The transformation from \(x^{\mu}\) to \(\bar{x}^{\mu}\) is given by forming the chain rule between them:
\( \frac{\partial \phi}{\partial \bar{x}^{\mu}} = (\frac{\partial}{\partial \bar{x}^{\mu}}) ( \frac{\partial x^{\nu}}{\partial x^{\nu}}) \phi \)
\( \frac{\partial \phi}{\partial \bar{x}^{\mu}} = \underset{\nu}{\Sigma} (\frac{\partial x^{\nu}}{\partial \bar{x}^{\mu}}) \frac{\partial \phi}{\partial x^{\nu}} \) (Covariant Transformation)
In the usual Euclidean situation there is no practical difference between the two kinds of vectors, but in more general multilinear algebra the covariants are related to dual spaces and differential forms. For our purposes "upstairs" and "downstairs" vectors form inner products, which means they map to the reals (the products change the tensor rank.)
Einstein Summation Convention
The summation convention suppresses the \( \Sigma \) terms by assuming repeated indices, in this case \( \nu \), meaning they are summed:
Contravariant: \( \bar{a}^{\mu} = ( \frac{\partial \bar{x}^{\mu}}{\partial x^{\nu}} ) a^{\nu} \) (Equation 3)
Covariant: \( \partial_{\bar{\mu}} = \frac{\partial \phi}{\partial \bar{x}^{\mu}} = (\frac{\partial x^{\nu}}{\partial \bar{x}^{\mu}}) \frac{\partial \phi}{\partial x^{\nu}} \) (Equation 4)
Inner Products, Matrices, Tensors
Lorentz Transformation
The conversion of the x-axis Lorentz boosts, Equations 1 & 2, into matrix form:
$$\begin{bmatrix}
\bar{t} \\
\bar{x} \\
\bar{y} \\
\bar{z}
\end{bmatrix}
=
\begin{bmatrix}
\gamma & -\beta\gamma & 0 & 0 \\
-\beta\gamma & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
t \\
x \\
y \\
z
\end{bmatrix}
(Eqn. 1)
$$
$$\begin{bmatrix}
t \\
x \\
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
\gamma & \beta\gamma & 0 & 0 \\
\beta\gamma & \gamma & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
\bar{t} \\
\bar{x} \\
\bar{y} \\
\bar{z}
\end{bmatrix}
(Eqn. 2)
$$
The values of the middle matrix constrain the Lorentz transformation, where the symbol \(\Lambda\) is the coordinate transform:
\( \Lambda^{\mu}_{\phantom{\mu}\nu} = (\frac{\partial \bar{x}^{\mu}}{\partial x^{\nu}}) \)
\( \Lambda_{\phantom{\nu}\mu}^{\nu} = (\frac{\partial {x}^{\nu}}{\partial \bar{x}^{\mu}}) \)
Where their product is the Kronecker delta \(\delta^{j}_{i} \), identity being inverses of each other:
\( \Lambda^{\mu}_{\phantom{\mu}\nu} \Lambda_{\mu}^{\phantom{\mu}\rho} = (\frac{\partial \bar{x}^{\mu}}{\partial x^{\nu}}) (\frac{\partial {x}^{\rho}}{\partial \bar{x}^{\mu}}) = (\frac{\partial x^{\rho}}{\partial x^{\nu}}) = \delta^{\rho}_{\nu} \)
Such that \(\Lambda^{\mu}_{\phantom{\mu}\nu}\) transforms contravariant vectors, and its inverse \(\Lambda_{\mu}^{\phantom{\mu}\nu} \) transforms covariants:
\( \bar{x}^{\mu} = \Lambda^{\mu}_{\phantom{\mu}\nu}x^{\nu} \) (Eqn. 1)
\( \bar{x}_{\mu} = \Lambda_{\mu}^{\phantom{\mu}\nu}x_{\nu} \) (Eqn. 2)
Which in the index notation means the opposite "staired" indices convert the symbol of the vector index.
Inner Products
The Lorentz transformation leaves the length of the four-vector x unchanged in spacetime:
\(a \cdot b = a^{0}b^{0} - \overrightarrow{a} \cdot \overrightarrow{b} \)
\(x \cdot x = x^{0}x^{0} - \overrightarrow{x} \cdot \overrightarrow{x} \)
\(x \cdot x = (x^{0})^{2} - (x^{1})^{2} - (x^{2})^{2} - (x^{3})^{2} \)
Where the four-vector index values 0,1,2,3 are coordinates t,x,y,z.
Examples of Four-Vector Inner Products:
Energy-Momentum: \( p \cdot p = p_{\mu}^{\mu} = (E, \overrightarrow{p}) \cdot (E, \overrightarrow{p}) = E^{2} - \overrightarrow{p}^{2} = m_{0}^{2}c^{4} = m_{0}^{2} \)
Where \(m_{0}\) is the rest mass and c = 1 by convention, no implicit summation from \( \mu \) terms.
Derivative Operator: \( \partial_{\mu} x^{\nu} = \frac{\partial x^{\nu}}{\partial x^{\mu}} = \delta^{\nu}_{\mu} \rightarrow \partial_{\mu} x^{\mu} = 4 \)
Kronecker delta by definition, running over \(\mu\) index. Four dimensions, sums to 4.
d'Alembertian operator: \( \partial^{2} = \partial^{\mu} \partial_{\mu} = (\partial^{0})^{2} - \nabla^{2} \)
\( \Rightarrow \partial^{2} = \frac{\partial^{2}}{\partial t^{2}} - \nabla^{2} = \frac{\partial^{2}}{\partial t^{2}} - \frac{\partial^{2}}{\partial x^{2}} - \frac{\partial^{2}}{\partial y^{2}} - \frac{\partial^{2}}{\partial z^{2}} \)
Where the right hand side are normal partial derivatives in spite of the notational ambiguity. The d'Alembertian is the Laplace operator for Minkowski spacetime.
Tensors
Tensors are mathematical objects with arbitrary numbers of indices that preserve under coordination transformations. The metric tensor is the physical quantity that defines distance in spacetime. In the case of special relativity the symbol is usually \( \eta_{\mu \nu} \), which has a fixed value that never changes. More generally, you use \( g_{\mu \nu} \), which is situational in general relativity.
Minkowski Metric:
The Minkowski metric is what encodes the signature of the Lorentz transformation.
$$
\eta_{\mu\nu}
=
\begin{bmatrix}
1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 \\
0 & 0 & 0 & -1
\end{bmatrix}
$$
such that
\( a \cdot b = \eta_{\mu\nu}a^{\mu}b^{\nu} = a^{0}b^{0} - \overrightarrow{a}\cdot\overrightarrow{b} \)
\( a_{\mu} = \eta_{\mu\nu} a^{\nu} \)
which means we can raise and lower indices using the metric tensor.
Since the 3-vector portions are opposite signed, those upstairs and downstairs indices can be done either way:
\( a^{0} = a_{0} , a^{i} = -a_{i}, \eta_{\mu\nu} = \eta^{\mu\nu}\)
\( a \cdot b = \eta_{\mu\nu} a^{\mu} b^{\nu} = a_{\mu}b^{\mu} = a^{\mu}b_{\mu} = \eta^{\mu\nu}a_{\mu}b_{\nu} \)
General Tensor:
The transformation equation for a general tensor T is essentially gluing together the rules for vectors.
\( \bar{T}^{i' \cdots k'}_{l' \cdots n'} = (\frac{\partial \bar{x}^{i'}}{\partial x^{i}} \cdots \frac{\partial \bar{x}^{k'}}{\partial x^{k}})(\frac{\partial x^{l}}{\partial \bar{x}^{l'}} \cdots \frac{\partial x^{n}}{\partial \bar{x}^{n'}} ) T^{i \cdots k}_{l \cdots n} \)
where the Roman letters are not following the convention regarding spacetime dimensions.
Useful Operators:
* Operators are functions that map a space of physical states to another space of physical states. They are heavily used in more abstract treatments of physics, and are foundational in quantum mechanics, where quantum states are expressed into terms of classical observables. Classical variables such as position and momentum are only physical as eigenvalues.
* The Kronecker delta \( \delta^{i}_{j} \) is a mixed tensor of second rank. The rules are that i=j yields a value of 1, and every other case is a zero. When written as \(\delta_{ij}\) or \(\delta^{ij}\) they are scalars equaling 0 or 1 rather than tensors.
* The Levi-Civita symbol \( \epsilon^{ijkl} \) has useful antisymmetric properties, and will not be used as a tensor. This means that upper and lower indices will be treated as identical. It is related to determinants and cross products. The rules in four dimensions:
(1) \( \epsilon^{ijkl} = 1\) , for ijkl of 0123, if the number of times digits were moved is even (e.g. 2301, 4 moves)
(2) \( \epsilon^{ijkl} = -1\) , for ijkl of 0123, if the number of times digits were moved is odd (e.g. 0213, 1 move)
(3) \( \epsilon^{ijkl} = 0\) , for anything else (e.g. 0012 \( \Longrightarrow \epsilon^{0012} = 0\) )
The machinery of a function turns a number into some other number. Functionals turn functions into a number. Integral transforms turn functions into other functions. Integrating over a functional uses a range of functions instead of numbers. Generalized functions (also called distributions) reinterpret functions as functionals acting on functions, so that non-integrable functions can still be integrated.
Fourier Transforms
This is an integral transform essential for switching between spatial and frequency representations of quantum mechanics.
Fourier Transform: (\( \omega \) is the temporal component of the k 4-vector)
\( \tilde{f}(k) = \int \mathrm{d}^{4}x \hspace{0.1cm} \mathrm{e}^{(\mathrm{i} k \cdot x)} \hspace{0.1cm} f(x) \)
\( \tilde{f}(k) = \tilde{f}(\omega,\overrightarrow{k}) = \int \mathrm{d}^{3}x \hspace{0.1cm} \mathrm{d}t \hspace{0.1cm} \mathrm{e}^{(\omega t - \mathrm{i} \overrightarrow{k} \cdot \overrightarrow{x})} \hspace{0.1cm} f(t, \overrightarrow{x}) \)
Inverse Fourier Transform: (Factor of \( (2\pi)^{4} \) is the result of integrating four times)
\( f(x) = \int \frac{\mathrm{d}^{4}k}{(2\pi)^{4}} \hspace{0.1cm} \mathrm{e}^{(-\mathrm{i} k \cdot x)} \hspace{0.1cm} \tilde{f}(k)\)
Dirac Delta Function
This is a generalized function that gives the integral of even a discontinuous function a finite value, which is useful for things like point charges, where things go to infinity on the point but are zero immediately adjacent. It is frequently seen in field theory because of its relation to Green's functions, which are heavily used in propagators, and it is the inverse Fourier transform of constant numbers.
The Delta Function:
The integral of the delta function itself, in arbitrary d-dimensions, is equal to 1:
\( \int \mathrm{d}^{(d)} x \delta^{(d)} (x) = 1 \)
where, notably, the derivative of the Heaviside step function is the Dirac delta function.
Acting Upon Test Function:
Given a test function f(x), the integral of their product equals f(0), the value of f(x) at x=0:
\( \int \mathrm{d}^{(d)} f(x) \delta^{(d)}(x) = f(0) \)
Fourier Transform Cases:
Let \( f(x) = \mathrm{e}^{(\mathrm{i}k \cdot x)}\) for the case of the Fourier transform:
\( \tilde{\delta}^{(d)}(k) = \int \mathrm{d}^{(d)} \mathrm{e}^{(\mathrm{i}k \cdot x)} \delta^{(d)}(x) = f(0) = \mathrm(e)^{0} = 1 \)
The inverse Fourier transform of \( \tilde{f}(x) = 1\), therefore, is the Dirac delta function in d-dimensions:
\( \delta^{(d)} = \int \frac{\mathrm{d}^{(d)}k}{(2\pi)^{(d)}} \mathrm{e}^{(- \mathrm{i}k \cdot x)} \)
Green's Functions
These express inhomogeneous linear ordinary differential equations of some x(t), yielding the response f(t), as an operator acting on some "Green's function" G(t,u) which equals a set of Dirac delta functions. Since the function f(t) is the sum of such delta functions, the linearity of the operator means the solution to the differential equation is a sum of x(t), by integrating over the product \(G(t,u)f(t)\). It is essentially an integral transform method of solving ODEs. This is especially useful in impulse response problems in engineering, because of the delta function, and the fact that the "propagators" of single particles are the Green's functions of their (differential) equations of motion.
Green's Function Definition:
Let \( \hat{L} \) be some linear differential operator, and x(t) be some function, with the inhomogeneous ODE:
\( \hat{L} x(t) = f(t) \)
This can instead be expressed as generalized functions G(t,u) and Dirac delta functions over positions u:
\( \hat{L} G(t,u) = \delta (t - u) \)
The responses are the Dirac delta functions, so their sum is converted back, becoming the integral solution.
Proof:
\( \hat{L}x(t) = \int \mathrm{d}u \hspace{0.1cm} \hat{L} \hspace{0.1cm} G(t,u) \hspace{0.1cm} f(u) \)
\( \hspace{1.2cm} = \int \mathrm{d}u \hspace{0.1cm} \delta(t - u) \hspace{0.1cm} f(u) \)
\( \hspace{1.2cm} = f(t) \)
Poisson's Equation, Point Charge Case:
Poisson's equation in S.I. units is: \( \overrightarrow{\nabla}^{2} V(\overrightarrow{x}) = - \frac{\rho(\overrightarrow{x})}{\epsilon_{0}} \). Given a point charge whose magnitude for charge density \( \rho \) equals 1, localized as a peak at some position u, this is an almost trivial case where f(t) itself is the Dirac delta function:
\( \epsilon_{0} \overrightarrow{\nabla}^{2} V(\overrightarrow{x}) = - \delta^{3} (\overrightarrow{x} - \overrightarrow{u}) \)
Let the linear operator \( \hat{L} = \overrightarrow{\nabla}^{2} \) and the Green's function \( G(t,u) = V(\overrightarrow{x}) \):
\( \overrightarrow{\nabla}^{2} V(\overrightarrow{x}) = \hat{L} G(t,u) = - \delta^{3} (\overrightarrow{x} - \overrightarrow{u}) \)
From the laws of electromagnetism, the electrostatic potential \( V(\overrightarrow{x}) = \frac{1}{4\pi\epsilon_{0}\lvert \overrightarrow{x} - \overrightarrow{u} \rvert} \), so in Heaviside-Lorentz units:
\( \overrightarrow{\nabla}^{2} ( \frac{1}{4\pi \lvert \overrightarrow{x} - \overrightarrow{u} \rvert} ) = - \delta (\overrightarrow{x} - \overrightarrow{u}) \)
Given: \( \hat{L} G(t,u) = \delta(t - u) \)
\( \Rightarrow G(\overrightarrow{x},\overrightarrow{u}) = - \frac{1}{4\pi \lvert \overrightarrow{x} - \overrightarrow{u} \rvert }\)
Wavefunction Propagator:
The property that makes the Green's function approach most valuable for field theory is that it represents the "propagator" of integration that converts the values of the wave function \( \phi \) from one time and place to some other spacetime point:
Given \( \hat{L} x(t) = f(t)\) and \( \hat{L} G(t,u) = \delta(t - u) \):
\( \Rightarrow x(t) = \int \mathrm{d}u \hspace{0.1cm} G(t,u) \hspace{0.1cm} f(u) \)
\( \Rightarrow \phi(x, t_{x}) = \int \mathrm{d}y \hspace{0.1cm} G^{+}(x,t_{x},y,t_{y}) \hspace{0.1cm} \phi(y,t_{y}) \)
Where \( G^{+} = 0 \) for all times \( t_{x} < t_{y} \) so particles cannot travel into the past.
(1) "Quantum Field Theory for the Gifted Amateur"; Tom Lancaster & Stephen Blundell, Oxford University Press. p. 3-7 (2014)
(2) "Quantum Field Theory for the Gifted Amateur"; Tom Lancaster & Stephen Blundell, Oxford University Press. p. 144-146 (2014)
