Pseudovector

A loop of wire (black), carrying a current I, creates a magnetic field B (blue). If the position and current of the wire are reflected across the plane indicated by the dotted line, the magnetic field it generates would not be reflected: Instead, it would be reflected and reversed. The position of the wire and its current are vectors, but the magnetic field B is a pseudovector.^[1]

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, but in three dimensions gains an additional sign flip under an improper rotation such as a reflection. Geometrically it is the opposite, of equal magnitude but in the opposite direction, of its mirror image. This is as opposed to a true or polar vector, which on reflection matches its mirror image.

In three dimensions the pseudovector p is associated with the cross product of two polar vectors a and b:^[2]

\mathbf{p} = \mathbf{a}\times\mathbf{b}.\,

The vector p calculated this way is a pseudovector. One example is the normal to an oriented plane. An oriented plane can be defined by two non-parallel vectors, a and b,^[3] which can be said to span the plane. The vector a × b is a normal to the plane (there are two normals, one on each side – the right-hand rule will determine which), and is a pseudovector. This has consequences in computer graphics where it has to be considered when transforming surface normals.

A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. In mathematics pseudovectors are equivalent to three-dimensional bivectors, from which the transformation rules of pseudovectors can be derived. More generally in n-dimensional geometric algebra pseudovectors are the elements of the algebra with dimension n − 1, written Λⁿ⁻¹Rⁿ. The label 'pseudo' can be further generalized to pseudoscalars and pseudotensors, both of which gain an extra sign flip under improper rotations compared to a true scalar or tensor.

Physical examples

Physical examples of pseudovectors include magnetic field, torque, vorticity, and the angular momentum.

Each wheel of a car driving away from an observer has an angular momentum pseudovector pointing left. The same is true for the mirror image of the car.

Consider the pseudovector angular momentum L = r × p. Driving in a car, and looking forward, each of the wheels has an angular momentum vector pointing to the left. 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, corresponding to the extra minus sign in the reflection of a pseudovector.

The distinction between vectors and pseudovectors becomes important in understanding the effect of symmetry on the solution to physical systems. Consider an electric current loop in the z = 0 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.

Details

The definition of a "vector" in physics (including both polar vectors and pseudovectors) is more specific than the mathematical definition of "vector" (namely, any element of an abstract vector space). Under the physics definition, a "vector" is required to have components that "transform" in a certain way under a proper rotation: In particular, if everything in the universe were rotated, the vector would rotate in exactly the same way. (The coordinate system is fixed in this discussion; in other words this is the perspective of active transformations.) Mathematically, if everything in the universe undergoes a rotation described by a rotation matrix R, so that a displacement vector x is transformed to x′ = Rx, then any "vector" v must be similarly transformed to v′ = Rv. This important requirement is what distinguishes a vector (which might be composed of, for example, the x-, y-, and z-components of velocity) from any other triplet of physical quantities (For example, the length, width, and height of a rectangular box cannot be considered the three components of a vector, since rotating the box does not appropriately transform these three components.)

(In the language of differential geometry, this requirement is equivalent to defining a vector to be a tensor of contravariant rank one.)

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 rotation matrix R, so that a position vector x is transformed to x′ = Rx. If the vector v is a polar vector, it will be transformed to v′ = Rv. If it is a pseudovector, it will be transformed to v′ = −Rv.

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

\mathbf{v}' = R\mathbf{v}

(polar vector)

\mathbf{v}' = (\det R)(R\mathbf{v})

(pseudovector)

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.

Behavior under addition, subtraction, scalar multiplication

Suppose v₁ and v₂ are known pseudovectors, and v₃ is defined to be their sum, v₃ = v₁ + v₂. If the universe is transformed by a rotation matrix R, then v₃ is transformed to

\mathbf{v_3}' = \mathbf{v_1}'+\mathbf{v_2}' = (\det R)(R\mathbf{v_1}) + (\det R)(R\mathbf{v_2}) = (\det R)(R(\mathbf{v_1}+\mathbf{v_2}))=(\det R)(R\mathbf{v_3}).

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, v₃ = v₁ + v₂. If the universe is transformed by a rotation matrix R, then v₃ is transformed to

\mathbf{v_3}' = \mathbf{v_1}'+\mathbf{v_2}' = (R\mathbf{v_1}) + (\det R)(R\mathbf{v_2}) = R(\mathbf{v_1}+(\det R) \mathbf{v_2}).

Therefore, v₃ is neither a polar vector nor a pseudovector. For an improper rotation, v₃ does not in general even keep the same magnitude:

|\mathbf{v_3}| = |\mathbf{v_1}+\mathbf{v_2}|,

but

|\mathbf{v_3}'| = |\mathbf{v_1}'-\mathbf{v_2}'|

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.)

Behavior under cross products

Under inversion the two vectors change sign, but their cross product is invariant [black are the two original vectors, grey are the inverted vectors, and red is their mutual cross product].

For a rotation matrix R, either proper or improper, the following mathematical equation is always true:

(R\mathbf{v_1})\times(R\mathbf{v_2}) = (\det R)(R(\mathbf{v_1}\times\mathbf{v_2}))

where v₁ and v₂ are any three-dimensional vectors. (This equation can be proven either through a geometric argument or through an algebraic calculation.)

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

\mathbf{v_3}' = \mathbf{v_1}' \times \mathbf{v_2}' = (R\mathbf{v_1}) \times (R\mathbf{v_2}) = (\det R)(R(\mathbf{v_1} \times \mathbf{v_2}))=(\det R)(R\mathbf{v_3}).

So v₃ is a pseudovector. Similarly, one can show:

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

Examples

From the definition, it is clear that a displacement vector is a polar vector. The velocity vector is a displacement vector (a polar vector) divided by time (a scalar), so is also a polar vector. Likewise, the momentum vector is the velocity vector (a polar vector) times mass (a scalar), so is a polar vector. Angular momentum is the cross product of a displacement (a polar vector) and momentum (a polar vector), and is therefore a pseudovector. Continuing this way, it is straightforward to classify any vector as either a pseudovector or polar vector.

The right-hand rule

Above, pseudovectors have been discussed using active transformations. An alternate approach, more along the lines of passive transformations, is to keep the universe fixed, but switch "right-hand rule" with "left-hand rule" everywhere in math and physics, including in the definition of the cross product. Any polar vector (e.g., a translation vector) would be unchanged, but pseudovectors (e.g., the magnetic field vector at a point) would switch signs. Nevertheless, there would be no physical consequences, apart from in the parity-violating phenomena such as certain radioactive decays.^[4]

Formalization

One way to formalize pseudovectors is as follows: if V is an n-dimensional vector space, then a pseudovector of V is an element of the (n−1)-st exterior power of V: Λⁿ⁻¹(V). The pseudovectors of V form a vector space with the same dimension as V.

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 coincide, though it should be borne in mind that without additional structure (specifically, a volume form), there is no natural identification of Λⁿ⁻¹(V) with V.

Geometric algebra

In geometric algebra the basic elements are vectors, and these are used to build a hierarchy of elements using the definitions of products in this algebra. In particular, the algebra builds pseudovectors from vectors.

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:

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

where the leading term is the customary vector dot product and the second term is called the wedge product. 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 dimensions 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.^[5] In four dimensions, however, the pseudovectors are trivectors.^[6] In general, it is a (n − 1)-blade, where n is the dimension of the space and algebra.^[7] 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}.

Transformations in three dimensions

The transformation properties of the pseudovector in three dimensions has been compared to that of the vector cross product by Baylis.^[8] He says: "The terms axial vector and pseudovector are often treated as synonymous, but it is quite useful to be able to distinguish a bivector from its dual." To paraphrase Baylis: Given two polar vectors (that is, true vectors) a and b in three dimensions, the cross product composed from a and b is the vector normal to their plane given by c = a × b. Given a set of right-handed orthonormal basis vectors { e_ℓ }, the cross product is expressed in terms of its components as:

\mathbf {a} \times \mathbf{b} = (a^2b^3 - a^3b^2) \mathbf {e}_1 + (a^3b^1 - a^1b^3) \mathbf {e}_2 + (a^1b^2 - a^2b^1) \mathbf {e}_3 ,

where superscripts label vector components. On the other hand, the plane of the two vectors is represented by the exterior product or wedge product, denoted by a ∧ b. In this context of geometric algebra, this bivector is called a pseudovector, and is the dual of the cross product.^[9] The dual of e₁ is introduced as e₂₃ ≡ e₂e₃ = e₂ ∧ e₃, and so forth. That is, the dual of e₁ is the subspace perpendicular to e₁, namely the subspace spanned by e₂ and e₃. With this understanding,^[10]

\mathbf{a} \wedge \mathbf{b} = (a^2b^3 - a^3b^2) \mathbf {e}_{23} + (a^3b^1 - a^1b^3) \mathbf {e}_{31} + (a^1b^2 - a^2b^1) \mathbf {e}_{12} \ .

For details see Hodge dual. Comparison shows that the cross product and wedge product are related by:

\mathbf {a} \ \wedge \ \mathbf{b} = \mathit i \ \mathbf {a} \ \times \ \mathbf{b} \ ,

where i = e₁ ∧ e₂ ∧ e₃ is called the unit pseudoscalar.^[11]^[12] It has the property:^[13]

\mathit{i}^2 = -1 \ .

Using the above relations, it is seen that if the vectors a and b are inverted by changing the signs of their components while leaving the basis vectors fixed, both the pseudovector and the cross product are invariant. On the other hand, if the components are fixed and the basis vectors e_ℓ are inverted, then the pseudovector is invariant, but the cross product changes sign. This behavior of cross products is consistent with their definition as vector-like elements that change sign under transformation from a right-handed to a left-handed coordinate system, unlike polar vectors.

Note on usage

As an aside, it may be noted that not all authors in the field of geometric algebra use the term pseudovector, and some authors follow the terminology that does not distinguish between the pseudovector and the cross product.^[14] However, because the cross product does not generalize beyond three dimensions,^[15] the notion of pseudovector based upon the cross product also cannot be extended to higher dimensions. The pseudovector as the (n – 1)-blade of an n-dimensional space is not so restricted.

Another important note is that pseudovectors, despite their name, are "vectors" in the common mathematical sense, i.e. elements of a vector space. The idea that "a pseudovector is different from a vector" is only true with a different and more specific definition of the term "vector" as discussed above.

Notes

↑ Stephen A. Fulling; Michael N. Sinyakov; Sergei V. Tischchenko (2000). Linearity and the mathematics of several variables. World Scientific. p. 343. ISBN 981-02-4196-8.
↑ Aleksandr Ivanovich Borisenko; Ivan Evgenʹevich Tarapov (1979). Vector and tensor analysis with applications (Reprint of 1968 Prentice-Hall ed.). Courier Dover. p. 125. ISBN 0-486-63833-2.
↑ RP Feynman: §52-5 Polar and axial vectors from Chapter 52: Symmetry and physical laws, in: Feynman Lectures in Physics, Vol. 1
↑ See Feynman Lectures.
↑ William M Pezzaglia Jr. (1992). "Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations". In Julian Ławrynowicz. Deformations of mathematical structures II. Springer. p. 131 ff. ISBN 0-7923-2576-1.
↑ In four dimensions, such as a Dirac algebra, the pseudovectors are trivectors. Venzo De Sabbata; Bidyut Kumar Datta (2007). Geometric algebra and applications to physics. CRC Press. p. 64. ISBN 1-58488-772-9.
↑ William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓ_n: Duals". Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100. ISBN 0-8176-3257-3.
↑ William E Baylis (1994). Theoretical methods in the physical sciences: an introduction to problem solving using Maple V. Birkhäuser. p. 234, see footnote. ISBN 0-8176-3715-X.
↑ R Wareham, J Cameron & J Lasenby (2005). "Application of conformal geometric algebra in computer vision and graphics". Computer algebra and geometric algebra with applications. Springer. p. 330. ISBN 3-540-26296-2. In three dimensions, a dual may be right-handed or left-handed; see Leo Dorst; Daniel Fontijne; Stephen Mann (2007). "Figure 3.5: Duality of vectors and bivectors in 3-D". Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry (2nd ed.). Morgan Kaufmann. p. 82. ISBN 0-12-374942-5.
↑ Christian Perwass (2009). "§1.5.2 General vectors". Geometric Algebra with Applications in Engineering. Springer. p. 17. ISBN 3-540-89067-X.
↑ David Hestenes (1999). "The vector cross product". New foundations for classical mechanics: Fundamental Theories of Physics (2nd ed.). Springer. p. 60. ISBN 0-7923-5302-1.
↑ Venzo De Sabbata; Bidyut Kumar Datta (2007). "The pseudoscalar and imaginary unit". Geometric algebra and applications to physics. CRC Press. p. 53 ff. ISBN 1-58488-772-9.
↑ Eduardo Bayro Corrochano; Garret Sobczyk (2001). Geometric algebra with applications in science and engineering. Springer. p. 126. ISBN 0-8176-4199-8.
↑ For example, Bernard Jancewicz (1988). Multivectors and Clifford algebra in electrodynamics. World Scientific. p. 11. ISBN 9971-5-0290-9.
↑ Stephen A. Fulling; Michael N. Sinyakov; Sergei V. Tischchenko (2000). Linearity and the mathematics of several variables. World Scientific. p. 340. ISBN 981-02-4196-8.

General references

George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001). (ISBN 0-12-059815-9)
Chris Doran and Anthony Lasenby, Geometric Algebra for Physicists (Cambridge University Press: Cambridge, 2007) (ISBN 978-0-521-71595-9)
Richard Feynman, Feynman Lectures on Physics, Vol. 1 Chap. 52. See §52-5: Polar and axial vectors, p. 52-6
Axial vector at Encyclopaedia of Mathematics
J. D. Jackson, Classical Electrodynamics (Wiley: New York, 1999). (ISBN 0-471-30932-X)
Susan M. Lea, "Mathematics for Physicists" (Thompson: Belmont, 2004) (ISBN 0-534-37997-4)
William E Baylis (2004). "Chapter 4: Applications of Clifford algebras in physics". In Rafał Abłamowicz; Garret Sobczyk. Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100 ff. ISBN 0-8176-3257-3. : The dual of the wedge product a ∧ b is the cross product a × b.