# Why in the field theory, particle's motion is described by 0+1 dimensional field theory?

I started reading the lecture notes on Path integral formulation by Ashoke Das. At the very first page of the introduction chapter, he says that - "a theory describing the motion of a particle can be regarded as a special case, namely, we can think of such a theory as a $$(0+1)$$ dimensional field theory".

I don't understand why this is so. Maybe it is a very trivial question but I couldn't find a straightforward answer. I am puzzled by thinking why the time is enough to describe the motion? A particle surely has spatial dimension and can move in a 3-dimensional space as well. In absence of any external potential, a particle moves in a straight-line with constant velocity. This certainly requires a $$1+1$$ description. Doesn't it?

Field theory in $$(0+1)$$ dimensions is formally equivalent to particle mechanics.
Consider a scalar field $$\phi$$ in $$(d+1)$$ dimensions: it specifies a field value $$\phi(t, x_1,...,x_d)$$ for each point $$(t, x_1,...,x_d)$$ in spacetime. Therefore, it can be viewed as a mapping of spacetime into the real numbers: $$(t, x_1,...,x_d) \rightarrow \phi(t, x_1,...,x_d)$$.
The trajectory of a particle in $$(d + 1)$$ dimensions specifies a point in space for each moment of time. It can be viewed as a mapping of the real line (i.e. $$t \in \mathbb{R}$$) into space, namely: $$t \rightarrow (\phi_1(t), . . . , \phi_d(t))$$. I used the symbol $$\phi$$ to indicate the coordinated of the particle just to make more evident the analogy with the initial scalar field: now the "outcome" of the mapping are $$d$$ numbers but the initial ambient space is just a $$(0+1)$$ spacetime (i.e. only time). Loosely speaking: a collection of $$d$$ scalar fields in $$(0+1)$$ spacetime are equivalent to the motion of a particle in $$(d+1)$$ spacetime.
• Okay. So in particle mechanics, the objective is to find the spatial co-ordinates as a function of time. Thus the field for particle mechanics is actually the spatial co-ordinates. Or in other words, the field value $\phi$ is the value of the co-ordinates $(x,y,z)$ at some particular time $t$. Is this what you are trying to say? – Samapan Bhadury Oct 2 '20 at 13:22
• Yes, the coordinates of a particles are its "internal degrees of freedom", similarly to the $\phi$ value of the field, which is its internal degree of freedom. – Quillo Oct 2 '20 at 13:24