Concept of field in physics

Approximate image of a field

An abstract concept difficult to represent

The notion of field in physics is abstract and cannot be represented by an image. However, one can form a superficial idea of the concept of field. A field is a delimited portion of space filled with measurable physical quantities that can vary over time.

Scalar fields and vector fields

Example: in a room, at each point in space and at each instant, it is possible to measure physical quantities that characterize the state of the room, such as temperature, pressure, density, etc. The room is therefore filled with a temperature field, a pressure field, a density field, etc. These fields are said to be scalar because each point is represented by a real or complex number.

At each location in the room and at each instant, there is also a movement of air molecules that can be measured by two values, a direction and a speed. This velocity field is said to be vectorial because each location is represented by a quantity that has a direction and a magnitude. This quantity is a vector (an arrow). The length of the arrow represents the magnitude of the vector, in our case, the speed. The tip of the arrow represents the direction of the vector.

Examples of scalar and vector fields

The table below gathers some examples of scalar fields and vector fields observable in a room.

Representing a field: field lines and gradient

Field lines

To visualize a vector field, physicists draw field lines: curves tangent at each point to the field vector. Around a magnet, iron filings align along the magnetic field lines, making visible a normally invisible structure. Around an isolated electric charge, electric field lines radiate radially, denser where the field is intense.

The gradient of a scalar field

A scalar field is rather represented by equipotential surfaces (or lines), which connect points of the same value, somewhat like the contour lines of a topographic map. The gradient of this field indicates, at each point, the direction in which the quantity increases fastest: this is how a scalar field, such as temperature or electric potential, generates an associated vector field.

The field in classical physics

The analogy of the calm lake

If one wishes to represent a field by an approximate image, the simplest is to imagine a liquid expanse, a lake for example. At each location and at each instant, there is a movement of water molecules that can be measured by a vector value.

When the lake is in its state of equilibrium, that is, in its lowest energy state which can be called the ground state, nothing happens, it is perfectly calm. Each vector v = [x, y] has a zero component, that is, v = [0, 0]. The velocity field of the water molecules is immobile, empty, flat and invisible in a two-dimensional space.

The appearance of the field under the effect of wind

If the wind rises, an energy will be brought to the lake, the water molecules will become agitated and discernible ripples will form. Each velocity of a water molecule will become measurable by a non-zero vector. Each vector will represent a source of energy that generates an influence on the surrounding space.

While the field was empty, flat and invisible, it will manifest and appear in three dimensions. A physical field is then created. In other words, the physical field can be considered as a manifestation of the presence of a source of energy. The field does not directly represent the source but allows describing, in our case, how the physical properties of the lake vary at each point in space.

The field in quantum physics

The electromagnetic field

Fields in quantum physics are more complicated to imagine. The electromagnetic field is created by electric charges in motion, such as electrons, which are a source of energy. In this case, the field itself represents the distribution of electric and magnetic fields in space. The electromagnetic field is a combination of electric and magnetic fields that propagates through space in the form of electromagnetic waves.

The spinor field of electrons

The quantum field of electrons is spinorial: at each location and at each instant, one can measure the behavior of the electron's spin by a spinorial wave function. It is a complex mathematical function that characterizes the quantum state of the electron as a function of its position, time and spin. The quantum movements of the electron, such as its spin, also create magnetic fields. Spin is an intrinsic property of particles that is related to their magnetic moment.

Table of quantum fields

The table below summarizes the main quantum fields and their essential characteristics.

The quantum vacuum and its fluctuations

Superposition of fundamental fields

The quantum field of electrons, photons, protons, and all elementary particles, is an important theoretical concept for understanding the quantum properties of systems where all fundamental fields of particles superpose.

The quantum vacuum and fluctuations

When all these fields are in their ground state, of lowest energy, one is in the quantum vacuum, a calm sea filled with all particles without energy and undetectable. However, the quantum vacuum is not an "absolute nothingness", because it fluctuates permanently and gives birth to pairs of ephemeral particles called "virtual particle pairs".

Although these virtual particles exist only for an instant, their effects are not negligible: they are at the origin of phenomena measured experimentally, such as the Casimir effect (an attractive force between two very close metallic plates, due to vacuum fluctuations between them) or the Lamb shift, an infinitesimal displacement of the energy levels of the hydrogen atom.

The appearance of particles from the vacuum

These vacuum fluctuations can be interpreted as a quantum field that exists even in the absence of real particles. It will suffice to bring a source of energy to this quantum field for a ripple of the vacuum to emerge, a particle.

Measuring the gravitational field with cold atoms

The NASA QGGPf project

The concept of field is not just a theoretical abstraction: it also guides the design of measuring instruments. NASA's Jet Propulsion Laboratory (JPL) is developing the Quantum Gravity Gradiometer Pathfinder (QGGPf), a first space quantum sensor intended to map the Earth's gravitational field.

Cold atoms as test masses

The instrument uses two clouds of rubidium atoms cooled close to absolute zero: at this temperature, their behavior becomes wave-like, which allows measuring by quantum interference an infinitesimal difference in acceleration between the two clouds. This difference reflects local variations of the gravitational field, related to the distribution of masses beneath the Earth's surface (aquifers, mineral reserves, etc.). The launch of this mission, conducted with several industrial and academic partners, is envisaged around 2030.