Pseudoscalar

# Pseudoscalar

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In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a pseudoscalar is a quantity that behaves like a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

, except that it changes sign under a parity inversion
Parity (physics)
In physics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it is also commonly described by the simultaneous flip in the sign of all three spatial coordinates:...

such as improper rotation
Improper rotation
In 3D geometry, an improper rotation, also called rotoreflection or rotary reflection is, depending on context, a linear transformation or affine transformation which is the combination of a rotation about an axis and a reflection in a plane perpendicular to the axis.Equivalently it is the...

s while a true scalar does not.

The prototypical example of a pseudoscalar is the scalar triple product. A pseudoscalar, when multiplied by an ordinary vector
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

, becomes a pseudovector
Pseudovector
In physics and mathematics, a pseudovector is a quantity that transforms like a vector under a proper rotation, but 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...

or axial vector; a similar construction creates the pseudotensor
Pseudotensor
In physics and mathematics, a pseudotensor is usually a quantity that transforms like a tensor under an orientation preserving coordinate transformation , but gains an additional sign flip under an orientation reversing coordinate transformation In physics and mathematics, a pseudotensor is usually...

.

Mathematically, a pseudoscalar is an element of the top exterior power of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...

, or the top power of a Clifford algebra
Clifford algebra
In mathematics, Clifford algebras are a type of associative algebra. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal...

; see pseudoscalar (Clifford algebra). More generally, it is an element of the canonical bundle
Canonical bundle
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n is the line bundle\,\!\Omega^n = \omegawhich is the nth exterior power of the cotangent bundle Ω on V. Over the complex numbers, it is the determinant bundle of holomorphic n-forms on V.This is the dualising...

of a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...

.

## Pseudoscalars in physics

In physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, a pseudoscalar denotes a physical quantity
Physical quantity
A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...

analogous to a scalar
Scalar (physics)
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . This is in contrast to a vector...

. Both are physical quantities
Physical quantity
A physical quantity is a physical property of a phenomenon, body, or substance, that can be quantified by measurement.-Definition of a physical quantity:Formally, the International Vocabulary of Metrology, 3rd edition defines quantity as:...

which assume a single value which is invariant under proper rotations. However, under the parity transformation, pseudoscalars flip their signs while scalars do not. As reflection
Reflection (mathematics)
In mathematics, a reflection is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as set of fixed points; this set is called the axis or plane of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection...

s through a plane are the combination of a rotation with the parity transformation, pseudoscalars also change signs under reflections.

One of the most powerful ideas in physics is that physical laws do not change when one changes the coordinate system used to describe these laws. The fact that a pseudoscalar reverses its sign when the coordinate axes are inverted suggests that it is not the best object to describe a physical quantity. In 3-space, quantities which are described by a pseudovector are in fact skew-symmetric tensors of rank 3, which are invariant under inversion. The pseudovector is a much simpler representation of that quantity, but suffers from the change of sign under inversion. Specifically, in 3-space, the Hodge dual
Hodge dual
In mathematics, the Hodge star operator or Hodge dual is a significant linear map introduced in general by W. V. D. Hodge. It is defined on the exterior algebra of a finite-dimensional oriented inner product space.-Dimensions and algebra:...

of a scalar is equal to a constant times the 3-dimensional Levi-Civita pseudotensor
Levi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...

(or "permutation" pseudotensor); whereas the Hodge dual of a pseudoscalar is in fact a skew-symmetric (pure) tensor of rank three. The Levi-Civita pseudotensor is a completely skew-symmetric pseudotensor of rank 3. Since the dual of the pseudoscalar is the product of two "pseudo-quantities" it can be seen that the resulting tensor is a true tensor, and does not change sign upon an inversion of axes. The situation is similar to the situation for pseudovectors and skew-symmetric tensors of rank 2. The dual of a pseudovector is a skew-symmetric tensors of rank 2 (and vice versa). It is the tensor and not the pseudovector which is the representation of the physical quantity which is invariant to a coordinate inversion, while the pseudovector is not invariant.

The situation can be extended to any dimension. Generally in an N-dimensional space the Hodge dual of a rank n tensor (where n is less than or equal to N/2) will be a skew-symmetric pseudotensor of rank N-n and vice versa. In particular, in the four-dimensional spacetime of special relativity, a pseudoscalar is the dual of a fourth-rank tensor which is proportional to the four-dimensional Levi-Civita pseudotensor
Levi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...

.

### Examples

• the magnetic charge (as it is mathematically defined, regardless of whether it exists physically),
• the magnetic flux
Magnetic flux
Magnetic flux , is a measure of the amount of magnetic B field passing through a given surface . The SI unit of magnetic flux is the weber...

- it is result of a dot product
Dot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...

between a vector (the surface normal
Surface normal
A surface normal, or simply normal, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point P on the surface is a vector perpendicular to the tangent plane to that surface at P. The word "normal" is also used as an adjective: a line normal to a...

) and pseudovector (the magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...

),
• the helicity is the projection (dot product) of a spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...

pseudovector onto the direction of momentum
Momentum
In classical mechanics, linear momentum or translational momentum is the product of the mass and velocity of an object...

(a true vector).

## Pseudoscalars in geometric algebra

A pseudoscalar in a geometric algebra
Geometric algebra
Geometric algebra , together with the associated Geometric calculus, provides a comprehensive alternative approach to the algebraic representation of classical, computational and relativistic geometry. GA now finds application in all of physics, in graphics and in robotics...

is a highest-grade
Graded vector space
In mathematics, a graded vector space is a type of vector space that includes the extra structure of gradation, which is a decomposition of the vector space into a direct sum of vector subspaces.-N-graded vector spaces:...

element of the algebra. For example, in two dimensions there are two basis vectors, , and the associated highest-grade basis element is

So a pseudoscalar is a multiple of e12. The element e12 squares to −1 and commutes with all even elements — behaving therefore like the imaginary scalar i in the complex numbers. It is these scalar-like properties which give rise to its name.

In this setting, a pseudoscalar changes sign under a parity inversion, since if
→ (u1, u2)

is a change of basis representing an orthogonal transformation, then
e1e2u1u2 = ±e1e2,

where the sign depends on the determinant of the rotation. Pseudoscalars in geometric algebra thus correspond to the pseudoscalars in physics.