String-Theory and the String/Non-Local-‘Particles’ Duality

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 J in the string regime:




with the D-1-action:

    \[S_1^D = {\mkern 1mu} - {T_1}\int\limits_{{\rm{worldvolumes}}} {{d^{1 + 1}}} \xi \frac{{\not D_{\mu \nu }^{susy}L}}{{{{\not \partial }_{{v_a}}}}}{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{1 + 1}^{{\rm{array}}}} \right)}\]

and we get from:

    \[{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{1 + 1}^{{\rm{array}}}} \right)}\]

the d-1 mass-term:

    \[{T_1}{e^{ - {\Phi _{bos}}}}\prod\limits_{i = 1}^1 {\left( {2\pi nR} \right)} \]

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 \not Z' 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:

    \[{S^q} = - \int {dt} x\left( t \right)f\left( {{{\not \partial }_t}} \right)y\left( t \right)\]

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

    \[\begin{array}{l}\delta {S^q} = \int {dt\left\{ {\delta x\left[ {f\left( {{{\not \partial }_t}} \right)y\left( t \right)} \right]} \right.} + \\\delta y\left( t \right)\left[ {f\left( { - {{\not \partial }_t}} \right)x\left( t \right)} \right] + x\left( t \right)\left[ {f\left( {{{\not \partial }_t}} \right)\delta y\left( t \right)} \right] - \\\left. {\left[ {f\left( { - {{\not \partial }_t}} \right)x\left( t \right)} \right]\delta y\left( t \right)} \right\}\end{array}\]

with equations of motion:

    \[\left\{ {\begin{array}{*{20}{c}}{f\left( { - {{\not \partial }_t}} \right)x\left( t \right) = 0}\\{f\left( {{{\not \partial }_t}} \right)y\left( t \right) = 0}\end{array}} \right.\]

Solving for x and y gives the following solutions:

    \[x\left( t \right) = \sum\limits_a {{x_a}\,} {e^{ - i{k_a}t}}\]


    \[y\left( t \right) = \sum\limits_a {{y_a}\,} {e^{ - i{k_a}t}}\]

with \left\{ {{k_a}} \right\} being the set of zeros of f:

    \[\not {\rm Z}' = \left\{ {{k_a}} \right\} = \left\{ {k\left| {f\left( {ik} \right) = 0} \right.} \right\}\]

Thus, the total derivative of

    \[\begin{array}{l}\delta {S^q} = \int {dt\left\{ {\delta x\left[ {f\left( {{{\not \partial }_t}} \right)y\left( t \right)} \right]} \right.} + \\\delta y\left( t \right)\left[ {f\left( { - {{\not \partial }_t}} \right)x\left( t \right)} \right] + x\left( t \right)\left[ {f\left( {{{\not \partial }_t}} \right)\delta y\left( t \right)} \right] - \\\left. {\left[ {f\left( { - {{\not \partial }_t}} \right)x\left( t \right)} \right]\delta y\left( t \right)} \right\}\end{array}\]


    \[\begin{array}{c}x\left( t \right)\left[ {f\left( {{{\not \partial }_t}} \right)\delta y\left( t \right)} \right] - \left[ {f\left( { - {{\not \partial }_t}} \right)x\left( t \right)} \right]\delta y\left( t \right)\\ = \frac{d}{{dt}}\theta \end{array}\]

where \theta is:

    \[\theta = \bullet \left( {\frac{{1 \otimes f\left( {{{\not \partial }_t}} \right) - f\left( { - {{\not \partial }_t}} \right) \otimes 1}}{{1 \otimes {{\not \partial }_t} + {{\not \partial }_t} \otimes 1}}} \right)\left( {x\left( t \right) \otimes \delta y\left( t \right)} \right)\]


    \[ \bullet \left( {f \otimes g} \right) \equiv fg\]

So we get a Taylor expansion:

    \[f\left( {{{\not \partial }_t}} \right) = \sum\limits_{n = 0}^\infty {{f_n}} \not \partial _t^n\]

with the sympletic two-form

    \[\omega = \delta \theta \]


    \[\begin{array}{l}\omega = \bullet \left( {\frac{{1 \otimes f\left( {{{\not \partial }_t}} \right) - f\left( { - {{\not \partial }_t}} \right) \otimes 1}}{{1 \otimes {{\not \partial }_t} + {{\not \partial }_t} \otimes 1}}} \right)\left( {x\left( t \right) \otimes \delta y\left( t \right)} \right)\\ = \sum\limits_{n = 0}^\infty {{f_n}} \left\{ { \bullet \sum\limits_{k = 1}^n {\left[ {{{\left( { - {{\not \partial }_t}} \right)}^{k - 1}} \otimes \not \partial _t^{n - k}} \right]\left( {\delta x\left( t \right) \otimes \delta y\left( t \right)} \right)} } \right\}\end{array}\]

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

    \[\omega = \sum\limits_a {\dot f} \left( {i{k_a}} \right)\delta {x_a}\delta {y_a}\]

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

    \[x\left( t \right) \to x'\left( t \right) = x\left( {t + \delta t} \right)\]

In the harmonic basis:

    \[\left\{ {\begin{array}{*{20}{c}}{\delta t = \varepsilon {e^{{\mathop{\rm i}\nolimits} n\,t}} + c.c.}\\{\forall n \in \mathbb{Z}}\end{array}} \right.\]

c.c. being complex conjugation and \varepsilon the infinitesimal transformation of x generated by the differential operator:

    \[{V_n} = {e^{ - i\,n\,t}}{\not \partial _t}\]

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

    \[\left[ {{V_m},{V_n}} \right] = \left( {m - n} \right){V_{m + n}}\]

Now, compactifying t on a circle, I can normalize t so that t \in \left[ {0,2\pi } \right), thus {\left\{ {{e^{i\,n\,t}}} \right\}_{n \in \mathbb{Z}}} constitutes a complete basis for functions of t and the Virasoro algebra is equivalent to the full reparametrization symmetry of t.

Hence, a length scale is injected corresponding to the shift \Delta t in t that normalizes to 2\pi, 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:

    \[\left\{ {\begin{array}{*{20}{c}}{x\left( t \right)\sum\limits_{k \in \not {\rm Z}} {{x_k}{e^{ - ikt}}} }\\{y\left( t \right)\sum\limits_{k \in \not {\rm Z}} {{y_k}{e^{ - ikt}}} }\end{array}} \right.\]

If f is symmetric, we have

    \[f\left( { - {{\not \partial }_t}} \right) = f\left( {{{\not \partial }_t}} \right)\]

and {z_i} always comes in pairs \left( {{z_i} - {z_i}} \right) except for {z_i} = 0 or \frac{1}{2}. Hence, the symplectic two-form:

    \[\omega = \frac{1}{2}\ddot f\left( {i{k_a}} \right)\left( {\delta {x_a}\delta {q_a} + \delta {y_a}\delta {p_a}} \right) + ...\]


    \[\omega = \sum\limits_{k \in \not {\rm Z}'} {i\left( {k + n} \right)} \dot f\left( {ik} \right){x_k}{y_{\left( {k + n} \right)}}\]

with the conserved charges \theta with \delta x, \delta y given by the symmetry transformation:

    \[{Q_n} = \sum\limits_{k \in \not {\rm Z}'} {i\left( {k' + n} \right)} \dot f\left( {ik'} \right){x_k}{y_{\left( {k' + n} \right)}}\]

with the creation/annihilation operators:

    \[\left\{ {\begin{array}{*{20}{c}}{{a_{ - k}} = {{\left( {\dot f\left( {ik} \right)k} \right)}^{1/2}}{x_k}}\\{{b_k} = {{\left( {\dot f\left( {ik} \right)k} \right)}^{1/2}}{y_k}}\end{array}} \right.\]

Therefore, the symplectic two-form is equivalent to

    \[\omega = \sum\limits_{k \in \not {\rm Z}'} {\frac{{\delta {a_{ - k}}\delta {b_k}}}{k}} \]

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

    \[{Q_n} = \sum\limits_{k \in \not {\rm Z}'} {B\left( {n,k} \right)} \,{a_{ - k}}{b_{k + n}}\]


    \[B\left( {n,k} \right) = {\left( {\frac{{\dot f\left( {ik} \right)}}{{\dot f\left( {i\left( {k + n} \right)} \right)}}\frac{{k + n}}{k}} \right)^{1/2}}\]

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:

    \[\left\{ {\begin{array}{*{20}{c}}{{Q_m} \to U{Q_m}{U^{ - 1}}}\\{U = {e^{{\Sigma _k}{\lambda _k}{a_{ - 1}}{b_k}}}}\end{array}} \right.\]

with the following holding:

    \[B\left( {n,k} \right) \to \frac{{{e^{k{\lambda _k}}}}}{{{e^{\left( {n + k} \right){\lambda _{\left( {n + k} \right)}}}}}}B\left( {n,k} \right)\]

and since:

    \[{\lambda _k} \equiv \frac{1}{{2k}}\log \left[ {k/\dot f\left( {ik} \right)} \right]\]

I can deduce:

    \[B\left( {n,k} \right) \to 1\]

Summary so far: for the action

    \[{S^q} = - \int {dt} x\left( t \right)f\left( {{{\not \partial }_t}} \right)y\left( t \right)\]

non-local particle theory with conformal symmetry is characterized by its spectrum \not {\rm Z}' in string-theory