## On the concept of scalar field

In physics (cosmology)

“…scalar fields—fields that look the same no matter how you view them, but can contain energy or pressure. Their high level of symmetry suggests that one would be most likely to find them in the earliest moments of the universe’s history, which makes them relevant to cosmology. And scalar fields have the special property that they can have negative pressure—a curiosity that takes on deeper meaning in the context of general relativity, where pressure contributes to the curvature of spacetime. Normal pressure produces gravity. Negative pressure produces inflation.” – Amanda Gefter

In mathematics and physics

“In mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number or a physical quantity. Scalar fields are required to be coordinate-independent, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.” – Wikipedia

• A “scalar value” is not necessarily a real number. A “scalar value” can be a complex number as well.
• A (scalar, vector, tensor) field has infinite degrees of freedom because a field pervades all space by definition and each point in space can have any number(s)  as the field value(s).
• The gradient of a scalar field is a vector field
• Line integral of a scalar field is independent of path direction

I am a fan of the Khan Academy. Sal Khan has done great service by creating quality educational content on the web. I could not resist mentioning this educational video by Sal Khan where he proves that the line integral of a scalar field is independent of path direction. As you can guess, he also prepared another video where he shows that line integrals over vector fields are path direction dependent.

• In QFT (quantum field theory) a field ϕ is a distribution of quantum operators.
• In QFT a field ϕ is not a wave function. In other words,  ϕ is not a quantum state, ϕ is  similar to the position observable (operator) in quantum mechanics.