One of the deepest result in quantum physics is that the Wave-Particle-Duality-Relations correspond to a modern formulation of the Heisenberg uncertainty principle stated in terms of entanglement entropies: a Bell-type argument in the context of T-duality’s winding modes, shows the action describing any particle implies an ontological non-locality intrinsic to the notion of ‘particle’ and in this series of posts, I will show how that entails that ‘non-local-particles‘ (Pages 51-67) can be symmetrically identified with strings in string-theory. Consider, as I showed, the non-commutative action for the dynamics of N-branes of type in the string regime:

with the D-1-action:

and we get from:

the d-1 mass-term:

noting that generalizations to fields and D/p-branes are straightforward. In this, part one, I will draw a summary showing that non-local particle-theory with conformal symmetry is characterized by its spectrum in string-theory.

#### Nonlocal actions of particles derived via Feynman integral decomposition on a conformal hypersurface with conformal symmetry gives us a Virasoro algebra as guaranteed by Noether’s theorem

##### Let us look at what string-theory says about their canonical quantization

Consider quadratic actions of the form:

with scalar fields and . A canonical quantization yields the variational action:

with equations of motion:

Solving for and gives the following solutions:

and

##### with being the set of zeros of :

Thus, the **total derivative** of

is:

where is:

with

So we get a Taylor expansion:

with the sympletic two-form

being:

##### Since all zeros of are non-degenerate, we get:

#### Now, there are nonlocal particle actions satisfying conformal symmetry as a realized symmetry of string-theory worldline reparametrization

In the harmonic basis:

being complex conjugation and the infinitesimal transformation of generated by the differential operator:

and this is the crucial point: it satisfies the Virasoro algebra

Now, compactifying on a circle, I can normalize so that , thus constitutes a complete basis for functions of and the Virasoro algebra is **equivalent** to the full reparametrization symmetry of .

Hence, a length scale is injected corresponding to the shift in that normalizes to , with the Virasoro algebra generators constraining the normalization, the non-locality of particles has the gauge symmetry of the conformal group intrinsic to string theory

Solving, we get the equations of motion:

If is symmetric, we have

and always comes in pairs except for or . Hence, the symplectic two-form:

becomes:

with the conserved charges with , given by the symmetry transformation:

##### with the creation/annihilation operators:

### Therefore, the symplectic two-form is equivalent to

Important: the Virasoro algebraic constraints eliminate ghosts since the conserved charges are:

with:

with non-negative spectrum upon quantization after renormalization-ordering, and so the Virasoro algebra is satisfied at the semi-classical tree-level, as evidenced by:

with the following holding:

and since:

I can deduce:

## Summary so far: for the action

non-local particle theory with conformal symmetry is characterized by its spectrum in string-theory

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