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Scalars

Scalars are usually real or complex numbers, but other scalars may be used. In pure mathematics, a structure obeying the rules of arithmetic is a field. This is distinct from a scalar field, which means a scalar valued function on a coordinate system or a manifold. To avoid confusion, I will not use the word field in the algebraic sense, and will instead refer to sets of scalars, in which the rules of addition and multiplication are implicit.

Vector Space

A vector space is any set of objects (vectors) which can be added and multiplied by scalars to get another vector.


One can show that there is a zero vector, 0, with x + 0 = x for any vector x, and that, for every vector x, there is an additive inverse, –x, with x + –x = 0. Rules 1 - 5 define what is meant by vector space and lie at the heart of linear algebra.


I am going to describe only finite dimensional vector spaces. That is sufficient for empirical science, together with the idea that we can make the dimension, n, so large that making it any larger makes no practical difference to answers. Strictly speaking, that is also all that is justified by scientific empiricism. Mathematicians also talk about infinite dimensional vector spaces, but using them introduces difficulties, and the possibity of false proofs containing subtle errors which are difficult to spot. When done to the standards of rigour of pure mathematics, there is no chance of such an error, but those standards are not generally applied in either theoretical or experimental physics.

Coordinates

The sense of this definition of dimension can be seen by representing the elements of a basis as n × 1 column matrices, each with a 1 in one slot and a 0 in all others. Then the members of any vector space can be written simply as n × 1 column matrices. The values in each slot of the matrix are the coordinates of the vector. A vector can thus be specified by its coordinates in a given basis. As a shorthand, we often write a vector in terms of coordinates, using index notation, x = xⁱ.

The components of a vector are also vectors, represented as column matrices with xⁱ in the ith slot, and 0 in each other slot. For example, an ordinary 3-vector, or space vector, x, can be written as a sum of its components in a coordinate system with axes labelled 1, 2, and 3:
In Cartesian coordinates the components are at right angles to each other.

By writing vectors as column matrices, we can see that all vectors behave just like ordinary space vectors in three dimensions. The only difference is the number of slots in the matrix representation, i.e. the dimension. It’s useful to remember this when working with abstract vectors in n dimensions, because their properties can be understood by thinking of vectors in two or three dimensions.

The Dot Product

In two or three dimensions, the dot product gives a measure of the angle between vectors. It is found by projecting one vector onto another and multiplying the resulting magnitudes. For two vectors x and y, with magnitudes |x| and |y|, placed base to base so that the angle between them is θ, the dot product is The dot product depends only on the magnitudes of the two vectors and the angle between them, not on the coordinate system. It is an invariant.

Ket Notation

For 3-vectors and 4-vectors, it is normal to use index notation. Dirac introduced ket notation for treating the vectors used in quantum mechanics. In ket notation a vector is written . Kets are formed from a basis of kets representing states of particles with definite position. Thus, is a vector describing the state of a particle at position x. In standard quantum theory, there is a mathematical difficulty because it is assumed that there are an infinite number of positions where a particle could be. In relational quantum gravity, position means position with respect to a particular measurement apparatus. Any measurement apparatus only has a finite range and resolution, so can only determine a finite, albeit very large, number of positions. Relational quantum gravity uses finite dimensional vector space. You can get back to the standard theory by letting the number of dimensions go to infinity, so long as you don’t let subtle errors creep in. Kets can be thought of as N × 1 matrices where N is a very large number. For a particle at position x, the ket would be represented as a matrix with a 1 in the slot corresponding to position x and a 0 in all other slots.

Ket notation has simple and powerful features and is very elegant. Just as all vectors can be thought of as n × 1 matrices, all vectors can be described using ket notation. From here on, I will use ket notation to describe vectors in any vector space, not just in quantum theory. Then I will translate ket notation back into coordinate notation for dealing with vectors in space and spacetime. This will kill two birds with one stone; it will lay down the foundation for understanding quantum mechanics and at the same time treat vectors in a completely general way, so that you can see how exactly the same mathematics applies to completely different situations. In ket notation, the 3-vector x would be written:

Hilbert Space

The dot product is used for 3-vectors and 4-vectors. More generally a product between vectors with the same mathematical properties is an inner product. A vector space with an inner product is an inner product space. I will use the term Hilbert space. There is a lovely story that when Hilbert space was discussed at talk in a conference, a mathematician put his hand up and interrupted. “What is this Hilbert space?” he asked. The mathematician was David Hilbert. Originally Hilbert space was infinite dimensional, but now the term is also used for finite dimensional spaces. For finite dimensional vector spaces there is no difference between an inner product space and a Hilbert space. An inner product is a mapping which takes any two vectors to a scalar, which is linear in the second vector, and which is such that reversing the vectors is complex conjugation.


Taken together, rules 1 - 7 imply that we only need to know the inner product for basis elements in order to calculate it for any pair of vectors. For real vectors (i.e. vectors in a space using real numbers as scalars), rule 6 implies . Strictly, an extra condition is required for an inner product, that the inner product of a non-zero vector , with itself is always positive, , but this is not true for the products used in relativity. To avoid unnecessary linguistic complication, I will use the term inner product even when this condition is not satisfied, and I will talk of a positive definite inner product if I wish to assert that is.


Strictly a norm should satisfy , but this condition is not satisfied in relativity. Again, to avoid unnecessary language, I will not use it. Instead I will say that the norm is positive definite when, for all non-zero vectors, , we have .

Dirac’s Joke

Professor Polkinghorne, who taught me quantum theory, was very fond of telling Dirac’s joke. Apparently, when Dirac himself taught the subject, much of which was his own development, the only time he displayed any personal pride or pleasure in his achievement was when he told the joke: is a bra, and is a braket.


Using rules 6 and 7 it is easy to see that bras obey the rules of vector space (rules 1 -5). So, bras are also vectors. Because there is a one-one correspondence between bras and kets, they can often be regarded as different representations of the same vector. In quantum theory this is known as the state vector.

The Cauchy-Schwarz Inequality

Proof:  Trivial if . Let λ be a scalar. Then
Choose
Then,
Taking square roots proves the Cauchy-Schwarz inequality, which will be used to prove the uncertainty principle.

The triangle inequality follows,
Take square roots:

The Dot Product as an Inner Product

It is easy to see using right angled triangles that the dot product of z with the sum of x and y is equal to the sum of the dot products of z with x and y. This works in three dimensions as well as two (imagine x coming out of the plane). It also works when x and y are multiplied by scalars. Thus, the dot product obeys linearity. So, the dot product is an inner product. This means that all the rules we can find for inner products apply straight away to the dot product.

Of course, if I was a category theorist, I probably wouldn’t be able to say that the dot product is an inner product. I would probably have to say that the dot product is a member of the category of inner products, or something like that. Then I would have to learn a wholeload more jargon before I could say anything at all. I think this is the sort of thing which happens when people are too clever for their own good, and try to approach mathematics as though it consists only of formal rules, rather than deductive thinking. All I can say is, thank heavens I am not a category theorist.

Orthonormal Bases

In three dimensions it is usual to work with unit vectors in three directions at right angles to each other. This is not strictly necessary. For 4-vectors in spacetime, Lorentz transformation caused the axes to become skewed, and in general relativity even more general coordinate systems are used. But it certainly makes life easier to use unit vectors at right angles to each other when we can. For general vector spaces we use the word orthogonal instead of right angles. This applies for positive definite inner products.


For orthonormal vectors ,
where the Kronecker delta is δ_xy = 0 if x ≠ y, and δ_xx = 1 (no summation, as both indices are at the bottom. See below). In an orthonormal basis the inner product can be used to find the components of a vector. Write the vector as a sum of its components in a basis with members, ,


Take the inner product with the basis ket
So, is the coefficient, f ^x. We have
Take the inner product with any vector ,
So, we have written the inner product as a sum of the products of the components of vectors and . This is just the same as multiplying a row matrix by a column matrix, so if we think of kets as column matrices, bras can be thought of as row matrices. This is true for any vectors and , so we can write a neat expression known as the resolution of unity,
The resolution of unity makes otherwise difficult calculations in wave mechanics really simple, and will be used repeatedly when we explore quantum theory.

Contravariant and Covariant Vectors

Different language is used for spacetime vectors, but the ideas are exactly the same. Vectors written with an index at the top, corresponding to kets, are called contravariant. Vectors in the dual space, corresponding to bras, are written with an index at the bottom, and are covariant. If a contravariant vector, xⁱ, is thought of as a column matrix, the corresponding covariant vector can be thought of as row matrix, with coordinates, x_i, defined so that the dot product is a simple matrix multiplication. There is a direct, one-one correspondence between contravariant and covariant vectors, and we usually think of xⁱ and x_i as contravariant and covariant representations of the vector x. The inner product, or dot product, in three dimensions is
In practice, products of vector quanties are determined by the sum of products of components, described in the formula. We are not bound by the representation as rows and columns.

Einstein Summation Convention

It is often said that Einstein was not a great mathematician, but if a mathematician is to be judged on original insights with a lasting benefit, I don’t see that this is so. Special relativity is based on just such a mathematical insight. So was the (re)discovery of teleparallelism which he tried to use in unified field theory and which is used in the teleconnection. Mathematics is often thought complicated, but, in my view, a true mathematician is one who sees how to make complicated problems simple. Einstein did this with the summation convention. He noticed that every time there is a repeated index it takes place inside a sum. This means it is not necessary to write the sum. Leaving it out makes many formulae far easier both to write and to read. Using the summation convention, the dot product is more simply written,
Henceforth, when an index is repeated, once as a suffix and once as a superfix, it will always mean that one sums over that index.

The Metric

Using an orthornormal basis the relationship between covariant and contravariant vectors is simple. In three dimensions,
We write down matrices g_ij and g^ij which take contravariant vectors to covariant vectors, and covariant vectors to contravariant vectors.
where the summation convention has been used for the index j. g_ij and g^ij are sometimes described as index lowering and raising operators. Later we will see that, just as x_i and xⁱ are covariant and contravariant representations of the same vector, x, g_ij and g^ij are covariant and contravariant representations of a tensor, g. In three dimensions,
The magnitude, |x|, of vector x is found from the dot product of x with itself and using Pythagoras’ theorem. It is
The summation convention has been used on both i and j. Up to this point, we have been using an orthonormal basis, represented by a right angled triad of unit vectors. In relativity we need to be able to talk about general coordinate systems. In the general case, the components, xⁱ, can be skewed and stretched, as happens, for example, to the coordinates in a map of the surface of the Earth on flat paper. The clever bit is that, however we distort the coordinate system, we always retain the relationship between contravariant and covariant vectors, defined from the inner product, and described in the simple equation:
Thus, vector magnitude is always given by
In this way, no matter how the components of vectors are stretched, g gives a measure of the true magnitude of vector quantities in a given neighbourhood. Because of this property, g is called the metric.

3 Dimensions plus Time

In Minkowski Coordinates, as used in special relativity, the momentum 4-vector, or 4-momentum, pⁱ = (p⁰, p¹, p², p³) is given by:
Then Pythagoras' theorem takes the form,
m is the magnitude of p and is invariant. We define p_i = (p₀, p₁, p₂, p₃) with
The index raising and lowering operators are
For 4-vectors, p and q,
is invariant under Lorentz transformation. Thus, we have the same formula for the dot product, Lorentz transformation is given by a matrix multiplication, and, in all respects apart from positive definite norm, 4-vectors in spacetime behave exactly like ordinary 3-vectors.

The Light Cone

Some treatments use a metric signature (–, +, +, +) to preserve the Euclidean metric in three dimensions. This is largely a matter of preference. I prefer to preserve the sign of energy, since vectors are more useful to describe motion than position and because I find this choice of signature natural in the description of antimatter. With the signature (+, –, –, –), real motions are described by vectors with real magnitudes, faster than light motions by vectors with imaginary magnitudes, and vectors into the past can naturally be assigned negative magnitudes. The invariant magnitude |A| = √A² of the spacetime vector A characterises it as:

Coordinate Transformation

Near the top of this page the components of a 3-vector, x, were shown in Cartesian Coordinates, labelled 1, 2, 3. Expressing x in terms of primed coordinates, labelled 1', 2', 3', is coordinate transformation. This can be done by writing x as a sum of its components, and by writing each component as a vector in the new primed coordinates. The result is equivalent to multiplying by a square matrix, where i' runs over the rows and j runs over the columns.
Interchanging the axes and the suffixes we have:
and hence (substituting x^i')
This is true for any vector, x. So,
is the the Kronecker delta, if j = l, and otherwise. Thus, is the inverse tranformation of .

The axes do not have to be at right angles. The formulae work just as well for skewed axes. General coordinate transformations, including rotation, reflection, shear and stretch, are all described by matrix multiplication in exactly the same way, and for any number of dimensions. Of course, we can only draw diagrams and visualise two or three dimensions. The power and elegance of the mathematics lies in knowing that that is all we ever have to do. The formulae are the same in any number of dimensions.

Let y ^j be a contravariant vector. y ^jx_j is an invariant. So,
is true for any values of y^i'. Hence the transformation law for a covariant vector is

Lorentz Transformation

The Lorentz transform is simply a coordinate transformation, and is represented by a matrix. Coordinate transformations do not affect the inner product. As a result they are represented by unitary matrices. Real unitary matrices are symmetrical. For a boost in the 1-direction the matrix representation is found by acting on 4-velocity such that 3-velocity becomes zero:

Spherical Coordinates

In spherical coordinates, the position coordinate, (r, θ, φ) is determined through radial distance r, zenith angle θ from a fixed axis, and azimuth angle φ from a second axis perpendicular to the first. By convention, radial distance is the 1-direction, zenith angle is the 2-direction and azimuth angle is the 3-direction. Small coordinate changes dr, dθ and dφ correspond to displacement vectors with magnitudes dr, rdθ and rsinθdφ. So, the metric in these coordinates is
Spherical coordinates for Minkowski spacetime also have 0-coordinate for time, and the metric has a (+, –, –, –) signature,

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