Pseudovector
In
One example of a pseudovector is the normal to an oriented
A number of quantities in physics behave as pseudovectors rather than polar vectors, including magnetic field and angular velocity. In mathematics, in three dimensions, pseudovectors are equivalent to 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 ⋀n−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
Consider the pseudovector angular momentum L = Σ(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 sign flip in the reflection of a pseudovector.
The distinction between polar vectors and pseudovectors becomes important in understanding
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. , 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
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
(In the language of
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
The discussion so far only relates to proper rotations, i.e. rotations about an axis. However, one can also consider
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.
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
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 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:
- .
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
Behavior under cross products
For a rotation matrix R, either proper or improper, the following mathematical equation is always true:
- ,
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
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
This is isomorphic to addition modulo 2, where "polar" corresponds to 1 and "pseudo" to 0.
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. Torque is angular momentum (a pseudovector) divided by time (a scalar), so is also a pseudovector. Continuing this way, it is straightforward to classify any of the common vectors in physics as either a pseudovector or polar vector. (There are the parity-violating vectors in the theory of weak-interactions, which are neither polar vectors nor pseudovectors. However, these occur very rarely in physics.)
The right-hand rule
Above, pseudovectors have been discussed using
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)-th exterior power of V: ⋀n−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 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 . Vectors transform in the fundamental representation of with data given by , so that for any matrix in , one has . Pseudovectors transform in a pseudofundamental representation , with . Another way to view this homomorphism for odd is that in this case . Then is a direct product of group homomorphisms; it is the direct product of the fundamental homomorphism on with the trivial homomorphism on .
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
where the leading term is the customary vector dot product and the second term is called the wedge product or exterior 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
Transformations in three dimensions
The transformation properties of the pseudovector in three dimensions has been compared to that of the
where superscripts label vector components. On the other hand, the plane of the two vectors is represented by the
For details, see Hodge star operator § Three dimensions. The cross product and wedge product are related by:
where i = e1 ∧ e2 ∧ e3 is called the
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 eℓ 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.[16] However, because the cross product does not generalize to other than three dimensions,[17] the notion of pseudovector based upon the cross product also cannot be extended to a space of any other number of dimensions. The pseudovector as a (n – 1)-blade in an n-dimensional space is not restricted in this way.
Another important note is that pseudovectors, despite their name, are "vectors" in the sense of being 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.
See also
- Exterior algebra
- Clifford algebra
- Antivector, a generalization of pseudovector in Clifford algebra
- Orientability — discussion about non-orientable spaces.
- Tensor density
Notes
- ISBN 981-02-4196-8.
- ^ "Details for IEV number 102-03-33: "axial vector"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-11-07.
- ^ "Details for IEV number 102-03-34: "polar vector"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-11-07.
- ^ a b c d RP Feynman: §52-5 Polar and axial vectors, Feynman Lectures in Physics, Vol. 1
- ^
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.
- ^ See Feynman Lectures, 52-7, "Parity is not conserved!".
- ^
William M Pezzaglia Jr. (1992). "Clifford algebra derivation of the characteristic hypersurfaces of Maxwell's equations". In Julian Ławrynowicz (ed.). Deformations of mathematical structures II. Springer. p. 131 ff. ISBN 0-7923-2576-1.
- ^
In four dimensions, such as a ISBN 978-1-58488-772-0.
- ^
William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓn: 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 978-0-12-374942-0.
- ^
Christian Perwass (2009). "§1.5.2 General vectors". Geometric Algebra with Applications in Engineering. Springer. p. 17. ISBN 978-3-540-89067-6.
- ^
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 978-1-58488-772-0.
- ^
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.
References
- Arfken, George B.; Weber, Hans J. (2001). Mathematical Methods for Physicists. Harcourt. ISBN 0-12-059815-9.
- Doran, Chris; Lasenby, Anthony (2007). Geometric Algebra for Physicists. Cambridge University Press. ISBN 978-0-521-71595-9.
- Feynman Lectures on Physics. Vol. 1. p. 52–6.
- Axial vector at Encyclopaedia of Mathematics
- ISBN 0-471-30932-X.
- Lea, Susan M. (2004). Mathematics for Physicists. Thompson. ISBN 0-534-37997-4.
- Baylis, William E (2004). "4. Applications of Clifford algebras in physics". In Abłamowicz, Rafał; Sobczyk, Garret (eds.). 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.
- Weinreich, Gabriel (1998), Geometrical Vectors, Chicago Lectures in Physics, The University of Chicago Press, p. 126, ISBN 9780226890487