Spacetime dimensions as bosons

[Kyleric]

Quantum mechanics in 1 + 1 dimensions is equivalent to qft in 0 + 1 dimensions. This is because the position $x(t)$ of a particle can be replaced by a scalar field $\phi(t)$, and the momentum is replaced by the momentum conjugate of $\phi(t)$.

Also, in the bosonic construction of heterotic string theory, the left moving bosons can be combined with the right moving ones to make out the coordinates of spacetime.

Does it mean that spacetime dimensions can be interpreted as bosonic fields, with an associated bosonic particle?
I know it's probably not the case so I'd like to know where the equivalence breaks down.

 

[eljose79]

Hello!

Your observation about the equivalence between quantum mechanics in 1+1 dimensions and quantum field theory in 0+1 dimensions is correct. This is due to the fact that in both cases, the position of a particle is replaced by a scalar field and the momentum is replaced by the momentum conjugate of that field. In other words, the dynamics of a single particle in 1+1 dimensions can be described by a field theory in 0+1 dimensions.

As for your question about interpreting spacetime dimensions as bosonic fields, there is some truth to that idea. In string theory, spacetime dimensions are indeed described by bosonic fields known as "background fields" which determine the geometry of the spacetime. However, it is important to note that these background fields are not associated with physical particles in the same way that the fields in quantum field theory are. They are simply mathematical descriptions of the spacetime itself.

The equivalence between quantum mechanics and quantum field theory in different dimensions is not absolute and there are certain limits where the equivalence breaks down. For example, in the case of heterotic string theory, the equivalence only holds for certain types of string theories and not all of them. Additionally, there are certain physical phenomena that can only be described by quantum field theory in higher dimensions and cannot be captured by quantum mechanics in lower dimensions.

I hope this helps clarify the concept of equivalence between different dimensions in quantum mechanics and quantum field theory. Keep exploring and asking questions!