Inner products of vectors

For a real or complex vector space \({V}\), we can generalize another Cartesian structure, the inner product (AKA scalar product, dot product). We define an inner product space as including a mapping from vectors to scalars denoted \({\left\langle v,w\right\rangle }\) (also denoted \({(v,w)}\) or \({v\cdot w}\)). The mapping must satisfy:

  • \({\left\langle v,w\right\rangle =\left\langle w,v\right\rangle ^{*}}\) where \({^{*}}\) denotes complex conjugation
  • \({\left\langle ax+y,bz\right\rangle =a^{*}b\left\langle x,z\right\rangle +b\left\langle y,z\right\rangle }\)
  • \({\left\langle v,v\right\rangle >0}\) except if \({v=0}\), in which case it vanishes

The first requirement implies that \({\left\langle v,v\right\rangle }\) is real, and that the inner product is symmetric for real scalars. The second requirement can be phrased as saying that the inner product is anti-linear (AKA conjugate-linear) in its first argument and linear in its second, or sesquilinear, and the first and second requirements together define a Hermitian form. A real inner product is then bilinear or multilinear, meaning linear in each argument. The third requirement above makes the inner product positive definite.

Δ Sometimes the definitions of both inner product and sesquilinear are reversed to make the second argument anti-linear instead of the first. This is sometimes called the “mathematics” convention, while ours would then be the “physics” convention.
Δ An inner product as we have defined it on a complex vector space is also called a Hermitian inner product, and a complex inner product space is sometimes called a Hermitian inner product space, Hermitian space, or unitary space.

Two vectors are defined to be orthogonal if their inner product vanishes. The orthogonal complement of a subspace \({W}\) of \({V}\) is the subspace of all vectors orthogonal to every vector in \({W}\), i.e. \({W^{\perp}\equiv\left\{ v\in V\mid\forall w\in W\left\langle v,w\right\rangle =0\right\} }\). A basis of \({W}\) together with a basis for its orthogonal complement \({W^{\perp}}\) forms a basis for all of \({V}\).

An inner product defines a special class of bases, the orthonormal bases \({\hat{e}_{\mu}}\) with \({\left\langle \hat{e}_{\mu},\hat{e}_{\nu}\right\rangle =\delta_{\mu\nu}}\) (\({\equiv1}\) if \({\mu=\nu}\), \({0}\) otherwise). If we then write \({v=v^{\mu}\hat{e}_{\mu}}\) and \({w=w^{\mu}\hat{e}_{\mu}}\), we have

\(\displaystyle \left\langle v,w\right\rangle =\sum_{\mu}v^{\mu*}w^{\mu}=v^{\dagger}w, \)

where in the first expression we take the complex conjugate of the components \({v^{\mu}}\), and the second is common in linear algebra, where we treat the vectors as column matrices of components, and the inner product is formed by matrix multiplication after taking the adjoint (hermitian conjugate) of the first matrix. Thus every finite-dimensional real or complex inner product space is isomorphic to \({\mathbb{R}^{n}}\) or \({\mathbb{C}^{n}}\) with their standard component inner products as above, and \({\left\langle v,v\right\rangle}\) yields the same real value when applied the complex vector \({v\in V}\) or the decomplexified real vector in \({V^{\mathbb{R}}}\).

The positive definite definition of a real inner product is sometimes relaxed to only require the product to be nondegenerate (AKA anisotropic), i.e. \({\left\langle v,w\right\rangle =0}\) for all \({w}\) only if \({v=0}\). We will instead refer to this type of form, a nondegenerate symmetric bilinear form, as a pseudo inner product (AKA pseudo-metric); it is characterized by the fact that \({\left\langle v,v\right\rangle }\) can be negative or vanish. A vector \({v}\) is called isotropic (AKA light-like) if \({\left\langle v,v\right\rangle =0}\). In the context of manifolds, an inner product is called a metric; and in the context of spacetime, variants of the inner product carry specific terminology. We will cover this terminology in the following.

A pseudo inner product does not yield a well-defined norm, but the real quantity \({\sqrt{\pm\left\langle v,v\right\rangle}}\) or the possibly imaginary quantity \({\sqrt{\left\langle v,v\right\rangle}}\) is nevertheless sometimes called the “length” of \({v}\). A pseudo inner product also defines orthonormal bases, with the definition modified to allow \({\left\langle \hat{e}_{\mu},\hat{e}_{\nu}\right\rangle =\pm\delta_{\mu\nu}}\). The number of positive and negative values turns out to be independent of the choice of orthonormal basis (a result known as Sylvester’s law of inertia), and this pair of integers \({\left(r,s\right)}\) is called the signature (although the order is sometimes reversed). We define \({\eta{}_{\mu\nu}\equiv \left\langle \hat{e}_{\mu},\hat{e}_{\nu}\right\rangle \equiv\eta^{\mu\nu}}\), so that it is \({\pm1}\) if \({\mu=\nu}\) (with \({r}\) positive values), \({0}\) otherwise. A general signature is called pseudo-Riemannian (AKA pseudo-Euclidean), a signature with \({s=0}\) is called Riemannian (AKA Euclidean), a signature with \({s=1}\) (or \({r=1}\)) is called Lorentzian (AKA Minkowskian), and the signature (3,1) is called Minkowskian (AKA Lorentzian). More specifically, the signature (3,1) is called the “mostly pluses” signature (AKA relativity, space-like, or east coast signature), while the signature (1,3) is called the “mostly minuses” signature (AKA particle physics, time-like, or west coast signature). The terms Euclidean and Minkowskian often imply an inner product \({\left\langle \hat{e}_{\mu},\hat{e}_{\nu}\right\rangle =\eta{}_{\mu\nu}}\) defined on the basis vectors of \({\mathbb{R}^{n}}\), which we denote \({\mathbb{R}^{r,s}}\) for an inner product of signature \({(r,s)}\), a common but not universal convention.

Δ The term “signature” also sometimes refers to the integer \({r-s}\).
Δ For an arbitrary basis, the number of positive and negative values of \({\left\langle e_{\mu},e_{\nu}\right\rangle}\) do not necessarily match the signature.

The vectors orthogonal to a light-like vector in a Lorentzian signature are scalar multiples of itself and the space-like vectors orthogonal to its space-like component. This shows that the concept of an orthogonal complement (which together with the original subspace comprises the entire space) is not applicable to pseudo inner products.

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