Pseudoscalars
A pseudoscalar is a physical quantity that remains invariant under rotations, like an ordinary scalar, but changes sign under a parity transformation, that is, under a spatial inversion.
In other words, a pseudoscalar is a scalar quantity that is sensitive to the orientation of space.
To understand what a pseudoscalar is, it is helpful to begin by clarifying the notion of a scalar.
The difference between a scalar and a pseudoscalar
A scalar is a quantity described by a single numerical value and is independent of direction. Its value remains unchanged even when the system is observed in a mirror configuration.
Typical examples of scalar quantities include mass, temperature, and energy.
Example. At a point (x,y) on a surface, the measured temperature is 22 degrees. If we invert the spatial coordinates ($ x \to -x $, $ y \to -y $), the temperature at the point (-x,-y) is still 22 degrees. In other words, even when the surface is viewed “in a mirror,” the temperature at that point remains unchanged.
A pseudoscalar, by contrast, changes sign under a spatial inversion:
\[ (x, y, z) \rightarrow (-x, -y, -z) \]
If a quantity has value \( S \), then after the reflection it becomes:
\[ S \rightarrow -S \]
Thus, although it is a scalar in the mathematical sense, a pseudoscalar depends on the orientation of space.
A concrete example
A fundamental example of a pseudoscalar is the scalar triple product:
\[ S = \vec a \cdot (\vec b \times \vec c) \]
The quantity \( S \) is a scalar from a mathematical point of view, but it changes sign under a spatial inversion.
This occurs because the cross product \( \vec b \times \vec c \) is a pseudovector, and taking its dot product with an ordinary vector \( \vec a \) produces a pseudoscalar.
Geometrically, this quantity represents the oriented volume of the parallelepiped spanned by the three vectors.
For example, consider three vectors in space:
\[ \vec a = (1,0,0) \]
\[ \vec b = (0,1,0) \]
\[ \vec c = (0,0,1) \]
The scalar triple product is:
\[ S = \vec a \cdot (\vec b \times \vec c) \]
First, compute the cross product:
\[ \vec b \times \vec c = (1,0,0) \]
Then compute the dot product:
\[ S = \vec a \cdot (1,0,0) \]
\[ S = (1,0,0) \cdot (1,0,0) = 1 \]
Thus, the value of the pseudoscalar is:
\[ S = 1 \]
Now apply a parity transformation, that is, a spatial inversion:
\[ (x,y,z) \to (-x,-y,-z) \]
The vectors become:
\[ \vec a' = (-1,0,0) \]
\[ \vec b' = (0,-1,0) \]
\[ \vec c' = (0,0,-1) \]
We now recompute the scalar triple product:
\[ S = \vec a' \cdot (\vec b' \times \vec c') \]
First, the cross product:
\[ \vec b' \times \vec c' = (1,0,0) \]
and therefore:
\[ S = \vec a' \cdot (1,0,0) \]
\[ S = (-1,0,0) \cdot (1,0,0) \]
\[ S = -1 \]
The numerical value changes sign after the spatial inversion, which is precisely the effect of the parity transformation.
\[ S = 1 \quad \longrightarrow \quad S' = -1 \]
This shows that the scalar triple product is a pseudoscalar: it behaves as a scalar under rotations but changes sign under spatial inversion, unlike a true scalar such as temperature.
Physical meaning of pseudoscalars
Pseudoscalars commonly appear in physics whenever spatial orientation plays a role, particularly in theories involving parity violation or quantities associated with chirality and left - right asymmetry.
And so on.
