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In mathematics and physics, a scalar field associates a single number (or scalar) to every point in space. Scalar fields are often used in physics, for instance to indicate the temperature distribution throughout space, or the air pressure.
Definition
A scalar field is a function
- <math> F(\mathbf{x}): \mathbb{R}^n \mapsto \mathbb{R}<math>
or
- <math> F(\mathbf{x}): \mathbb{R}^n \mapsto \mathbb{C}<math>
The scalar field can be visualized as a n-dimensional space with a real or complex number attached to each point in the space.
The derivative of a scalar field results in a vector field called the gradient.
Usage
- Potential field
- In quantum field theory a scalar field is associated with spin 0 particles, like mesons. The scalar field may be real or complex valued (depending on whether it will associate a real or complex number to every point of space-time). Complex scalar fields represent charged particles.
Other fields
- Vector fields, which associate a vector to every point in space. Some examples of vector fields include the electromagnetic field or the newtonian gravitational field.
- Tensor fields, which associate a tensor to every point in space. In general relativity, gravity is associated with a tensor field. In particular, with the riemann curvature tensor. In Kaluza-Klein_theory spacetime is extended to five dimensions and its riemann curvature tensor can be separated out into ordinary four-dimensional gravitation plus an extra set, which is equivalent to Maxwell's equations for the electromagnetic field, plus an extra scalar field known as the "dilaton".
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