Fields are [continous](Continuity.md) sets of quantities that stretch over _all space_ and _all time_. Thus fields exist in 3+1 dimensional [[Spacetime]] (if not otherwise restricted to 2+1 or 1+1 dimensions as in some toy models). At the most basic level at every point, a field is associated with a quantity, most generally denoted as $\varphi$. This quantity may be a: * [vector](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) if the field is a [vector field.](Vector%20fields.md) * A scalar, in a [[scalar field]]. * A [tensor](Tensors%20and%20Multilinear%20Algebra%20(index).md#Tensors), in a [[tensor field]]. # Properties of fields Fields are excited or perturbed at particular points and regions where the field exists, however, this excitation affects the field in other points and regions. And in-fact values associated with points on a field are correlated along space-time. Thus excitations at one point on the field affected the field as a whole. However, in order for a field to be _physical_ we must assume that $\varphi\rightarrow0$ at $\infty.$ A field modeling a phenomenon with an effect at infinity would be modeling a system with infinite energy and thus would be non-physical. # Quantized fields In [Mechanics](Mechanics%20(index).md) and extensions of mechanics like [general relativity](General%20relativity%20(index).md) we conceptualize fields as being something that interacts with _particles_ or more generally objects in space. However, in [quantum mechanics](Quantum%20Field%20Theory%20(Index).md), _every particle is described by a field_ - and we require a method of [quantizing field](Field%20Quantization.md)s in order to describe this in a quantum mechanical framework. ## Fields as correlated oscillators In order to construct a theory for [[Field Quantization]]s, as our starting point, we need to construct a model in which a field in [[Spacetime]] modeled at _every point_ by a [[Harmonic Oscillator]], forming an [oscillator field](Oscillator%20Field%20in%20Spacetime.md). #Mechanics/ClassicalFields #Mechanics/MathematicalFoundations