Pseudovector

 ( Linear Algebra ) 

Consider the pseudovector angular momentum . Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left (by the right-hand rule). If the world is reflected in a mirror which switches the left and right side of the car, the "reflection" of this angular momentum "vector" (viewed as an ordinary vector) points to the right, but the actual angular momentum vector of the wheel (which is still turning forward in the reflection) still points to the left (by the right-hand rule), corresponding to the extra sign flip in the reflection of a pseudovector.

The distinction between polar vectors and pseudovectors becomes important in understanding the effect of symmetry on the solution to physical systems. Consider an electric current loop in the plane that inside the loop generates a magnetic field oriented in the z direction. This system is symmetric (invariant) under mirror reflections through this plane, with the magnetic field unchanged by the reflection. But reflecting the magnetic field as a vector through that plane would be expected to reverse it; this expectation is corrected by realizing that the magnetic field is a pseudovector, with the extra sign flip leaving it unchanged.

In physics, pseudovectors are generally the result of taking the cross product of two polar vectors or the curl of a polar vector field. The cross product and curl are defined, by convention, according to the right hand rule, but could have been just as easily defined in terms of a left-hand rule. The entire body of physics that deals with (right-handed) pseudovectors and the right hand rule could be replaced by using (left-handed) pseudovectors and the left hand rule without issue. The (left) pseudovectors so defined would be opposite in direction to those defined by the right-hand rule.

While vector relationships in physics can be expressed in a coordinate-free manner, a coordinate system is required in order to express vectors and pseudovectors as numerical quantities. Vectors are represented as ordered triplets of numbers: e.g. $\backslash mathbf\{a\}=(a\_x,a\_y,a\_z)$, and pseudovectors are represented in this form too. When transforming between left and right-handed coordinate systems, representations of pseudovectors do not transform as vectors, and treating them as vector representations will cause an incorrect sign change, so that care must be taken to keep track of which ordered triplets represent vectors, and which represent pseudovectors. This problem does not exist if the cross product of two vectors is replaced by the exterior product of the two vectors, which yields a bivector which is a 2nd rank tensor and is represented by a 3×3 matrix. This representation of the 2-tensor transforms correctly between any two coordinate systems, independently of their handedness.

(In the language of differential geometry, this requirement is equivalent to defining a vector to be a tensor of contravariant rank one. In this more general framework, higher rank tensors can also have arbitrarily many and mixed covariant and contravariant ranks at the same time, denoted by raised and lowered indices within the Einstein summation convention.)

A basic and rather concrete example is that of row and column vectors under the usual matrix multiplication operator: in one order they yield the dot product, which is just a scalar and as such a rank zero tensor, while in the other they yield the dyadic product, which is a matrix representing a rank two mixed tensor, with one contravariant and one covariant index. As such, the noncommutativity of standard matrix algebra can be used to keep track of the distinction between covariant and contravariant vectors. This is in fact how the bookkeeping was done before the more formal and generalised tensor notation came to be. It still manifests itself in how the basis vectors of general tensor spaces are exhibited for practical manipulation.

The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also consider improper rotations, i.e. a mirror-reflection possibly followed by a proper rotation. (One example of an improper rotation is inversion through a point in 3-dimensional space.) Suppose everything in the universe undergoes an improper rotation described by the improper rotation matrix R, so that a position vector x is transformed to . If the vector v is a polar vector, it will be transformed to . If it is a pseudovector, it will be transformed to .

The transformation rules for polar vectors and pseudovectors can be compactly stated as

where the symbols are as described above, and the rotation matrix R can be either proper or improper. The symbol det denotes determinant; this formula works because the determinant of proper and improper rotation matrices are +1 and −1, respectively.

So v₃ is also a pseudovector. Similarly one can show that the difference between two pseudovectors is a pseudovector, that the sum or difference of two polar vectors is a polar vector, that multiplying a polar vector by any real number yields another polar vector, and that multiplying a pseudovector by any real number yields another pseudovector.

On the other hand, suppose v₁ is known to be a polar vector, v₂ is known to be a pseudovector, and v₃ is defined to be their sum, . If the universe is transformed by an improper rotation matrix R, then v₃ is transformed to

Therefore, v₃ is neither a polar vector nor a pseudovector (although it is still a vector, by the physics definition). For an improper rotation, v₃ does not in general even keep the same magnitude:

$|\backslash mathbf\{v\_3\}|\; =\; |\backslash mathbf\{v\_1\}+\backslash mathbf\{v\_2\}|,\; \backslash text\{\; but\; \}\; \backslash left|\backslash mathbf\{v\_3\}\text{'}\backslash right|\; =\; \backslash left|\backslash mathbf\{v\_1\}\text{'}-\backslash mathbf\{v\_2\}\text{'}\backslash right|$.

If the magnitude of v₃ were to describe a measurable physical quantity, that would mean that the laws of physics would not appear the same if the universe was viewed in a mirror. In fact, this is exactly what happens in the weak interaction: Certain radioactive decays treat "left" and "right" differently, a phenomenon which can be traced to the summation of a polar vector with a pseudovector in the underlying theory. (See parity violation.)

$(R\backslash mathbf\{v\_1\})\backslash times(R\backslash mathbf\{v\_2\})\; =\; (\backslash det\; R)(R(\backslash mathbf\{v\_1\}\backslash times\backslash mathbf\{v\_2\}))$,

$\backslash nabla\; \backslash times\; (R\backslash mathbf\{v\})\; =\; (\backslash det\; R)(R(\backslash nabla\; \backslash times\; \backslash mathbf\{v\}))$

Suppose v₁ and v₂ are known polar vectors, and v₃ is defined to be their cross product, . If the universe is transformed by a rotation matrix R, then v₃ is transformed to

$\backslash mathbf\{v\_3\}\text{'}\; =\; \backslash mathbf\{v\_1\}\text{'}\; \backslash times\; \backslash mathbf\{v\_2\}\text{'}\; =\; (R\backslash mathbf\{v\_1\})\; \backslash times\; (R\backslash mathbf\{v\_2\})\; =\; (\backslash det\; R)(R(\backslash mathbf\{v\_1\}\; \backslash times\; \backslash mathbf\{v\_2\}))\; =\; (\backslash det\; R)(R\backslash mathbf\{v\_3\}).$

• polar vector × polar vector = pseudovector
• pseudovector × pseudovector = pseudovector
• polar vector × pseudovector = polar vector
• pseudovector × polar vector = polar vector

Similarly, if v₁ is any known polar vector field and v₂ is defined to be its curl , then if the universe is transformed by the rotation matrix R, v₂ is transformed to

$\backslash mathbf\{v\_2\}\text{'}\; =\; \backslash nabla\; \backslash times\; \backslash mathbf\{v\_1\}\text{'}\; =\; \backslash nabla\; \backslash times\; (R\backslash mathbf\{v\_1\})\; =\; (\backslash det\; R)(R(\backslash nabla\; \backslash times\; \backslash mathbf\{v\_1\}))\; =\; (\backslash det\; R)(R\backslash mathbf\{v\_2\}).$

• ∇ × polar vector field = pseudovector field
• ∇ × pseudovector field = polar vector field

This definition is not equivalent to that requiring a sign flip under improper rotations, but it is general to all vector spaces. In particular, when n is even, such a pseudovector does not experience a sign flip, and when the characteristic of the underlying field of V is 2, a sign flip has no effect. Otherwise, the definitions are equivalent, though it should be borne in mind that without additional structure (specifically, either a volume form or an orientation), there is no natural identification of ⋀ ⁿ⁻¹( V) with V.

Another way to formalize them is by considering them as elements of a representation space for $\backslash text\{O\}(n)$. Vectors transform in the fundamental representation of $\backslash text\{O\}(n)$ with data given by $(\backslash mathbb\{R\}^n,\; \backslash rho\_\{\backslash text\{fund\}\},\; \backslash text\{O\}(n))$, so that for any matrix $R$ in $\backslash text\{O\}(n)$, one has $\backslash rho\_\{\backslash text\{fund\}\}(R)\; =\; R$. Pseudovectors transform in a pseudofundamental representation $(\backslash mathbb\{R\}^n,\; \backslash rho\_\{\backslash text\{pseudo\}\},\; \backslash text\{O\}(n))$, with $\backslash rho\_\{\backslash text\{pseudo\}\}(R)\; =\; \backslash det(R)R$. Another way to view this homomorphism for $n$ odd is that in this case $\backslash text\{O\}(n)\; \backslash cong\; \backslash text\{SO\}(n)\backslash times\; \backslash mathbb\{Z\}\_2$. Then $\backslash rho\_\{\backslash text\{pseudo\}\}$ is a direct product of group homomorphisms; it is the direct product of the fundamental homomorphism on $\backslash text\{SO\}(n)$ with the trivial homomorphism on $\backslash mathbb\{Z\}\_2$.

The basic multiplication in the geometric algebra is the geometric product, denoted by simply juxtaposing two vectors as in ab. This product is expressed as:

$\backslash mathbf\; \{ab\}\; =\; \backslash mathbf\; \{a\; \backslash cdot\; b\}\; +\backslash mathbf\; \{a\; \backslash wedge\; b\}\; \backslash \; ,$

where the leading term is the customary vector dot product and the second term is called the exterior algebra. Using the postulates of the algebra, all combinations of dot and wedge products can be evaluated. A terminology to describe the various combinations is provided. For example, a multivector is a summation of k-fold wedge products of various k-values. A k-fold wedge product also is referred to as a k-blade.

In the present context the pseudovector is one of these combinations. This term is attached to a different multivector depending upon the of the space (that is, the number of linearly independent vectors in the space). In three dimensions, the most general 2-blade or bivector can be expressed as the wedge product of two vectors and is a pseudovector.

In four dimensions, such as a Dirac algebra, the pseudovectors are multivector.

In general, it is a -blade, where n is the dimension of the space and algebra.

An n-dimensional space has n basis vectors and also n basis pseudovectors. Each basis pseudovector is formed from the outer (wedge) product of all but one of the n basis vectors. For instance, in four dimensions where the basis vectors are taken to be { e₁, e₂, e₃, e₄}, the pseudovectors can be written as: { e₂₃₄, e₁₃₄, e₁₂₄, e₁₂₃}.