Pseudovector

From Wikipedia, the free encyclopedia

In physics and mathematics, a pseudovector (or axial vector) is a quantity that transforms like a vector under a proper rotation, i.e. a transformation that rotates vectors and pseudovectors by an arbitrary angle about an arbitrary axis, but gains an additional sign flip under an improper rotation: a transformation that can be expressed as a proper rotation of a mirror image, or equivalently as an inversion through the origin followed by a proper rotation. The conceptual opposite of a pseudovector is a (true) vector or a polar vector (more formally, a contravariant vector).

A common way of constructing a pseudovector p is by taking the cross product of two (true) vectors a and b:

p = a × b.

Under inversion the two vectors change sign, but their cross product is invariant.

A simple example of an improper rotation in 3D (but not in 2D) is inversion through the origin: Each (true) vector v goes to −v. Under this transformation, a and b go to −a and −b (by the definition of a vector), but p clearly does not change. It follows that any improper rotation multiplies p by −1 compared to the rotation's effect on a true vector.

This concept 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.

Many occurrences of pseudovectors in mathematics and physics are more naturally analyzed as bivectors, following the calculus of differential forms; the double negation is natural for a bivector. However, bivectors are "less intuitive" in some senses than ordinary vectors, and since in R³ every bivector a ʌ b has a unique dual vector a × b, it is this dual which is more often used.

[edit] Physical examples

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

Often, the distinction between vectors and pseudovectors is overlooked, but it becomes important in understanding and exploiting the effect of symmetry on the solution to physical systems. For example, consider the case of an electrical current loop in the z=0 plane: this system is symmetric (invariant) under mirror reflections through the plane (an improper rotation), so the magnetic field should be unchanged by the reflection. But reflecting the actual magnetic field through that plane changes its sign—this contradiction is resolved by realizing that the mirror reflection of the field induces an extra sign flip because of its pseudovector nature.

Each wheel of this car driving into the page has an angular momentum psuedovector pointing left.

As another example, 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 points to the right, but the actual angular momentum vector of the wheel still points to the left, corresponding to the minus sign. This reflects the fact that the wheels are still turning forward. In comparison, the behaviour of a regular vector, like the position of the car, is quite different.

To the extent that physical laws would be the same if the universe were reflected in a mirror (equivalently, invariant under parity), the sum of a vector and a pseudovector is not meaningful. However, the weak force, which governs beta decay, does depend on the chirality of the universe, and in this case pseudovectors and vectors are added.

[edit] 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.) 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.

[edit] 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.)

[edit] Behavior under cross products

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, and is well known.)

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

[edit] 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.

[edit] Pseudovectors and 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" and vice-versa everywhere in physics, in particular 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.^[1]

[edit] References and footnotes

^ See Feynman.

Richard Feynman, Feynman Lectures on Physics, Vol. 1 Chap. 52. Web link.
George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists (Harcourt: San Diego, 2001). (ISBN 0-12-059815-9)
John David 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)
Chris Doran and Anthony Lasenby, "Geometric Algebra for Physicists" (Cambridge University Press: Cambridge, 2007) (ISBN 978-0-521-71595-9)

[edit] See also

Grassmann algebra

[0] See Feynman.

[1]

Pseudovector

From Wikipedia, the free encyclopedia

Contents

[edit] Physical examples

[edit] Details

[edit] Behavior under addition, subtraction, scalar multiplication

[edit] Behavior under cross products

[edit] Examples

[edit] Pseudovectors and the right-hand-rule

[edit] References and footnotes

[edit] See also

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