In De Broglie-Bohm Pilot-Wave Theory unique equations of motion and scalar fields for a particle can be formulated. This is done by finding a solution for a divergence free probability density current $\vec{J}(r,t)$ and then dividing by the Born Rule $|\vecΨ(r,t)|^{2}$ for velocity arising from Hamilton-Jacobi mechanics. This addition of divergence free probability current would still satisfy the Schrodinger continuity equation. It was found that for a 2D polar system a divergence free probability density is the gradient rotated by the matrix ( $S$ ) applied to a scalar field as such $S\vec{\nabla}φ(r)$ . A variety of equations of motion were used such that the dimensions of the equations are equivalent to a velocity, different scalar fields were then derived for a 2D quantum harmonic oscillator. It was found that for each unique velocity there was a unique scalar field associated with it, as such the equation of motion for a particle in De Broglie-Bohm pilot wave theory could be theoretically dependent on any arbitrary number of scalar fields. This ultimately means the equations of motion could be ambiguous as there are multiple mathematically valid solutions.In De Broglie-Bohm Pilot-Wave Theory unique equations of motion and scalar fields for a particle can be formulated. This is done by finding a solution for a divergence free probability density current $\vec{J}(r,t)$ and then dividing by the Born Rule $|\vecΨ(r,t)|^{2}$ for velocity arising from Hamilton-Jacobi mechanics. This addition of divergence free probability current would still satisfy the Schrodinger continuity equation. It was found that for a 2D polar system a divergence free probability density is the gradient rotated by the matrix ($S$) applied to a scalar field as such $S\vec{\nabla}φ(r)$. A variety of equations of motion were used such that the dimensions of the equations are equivalent to a velocity, different scalar fields were then derived for a 2D quantum harmonic oscillator. It was found that for each unique velocity there was a unique scalar field associated with it, as such the equation of motion for a particle in De Broglie-Bohm pilot wave theory could be theoretically dependent on any arbitrary number of scalar fields. This ultimately means the equations of motion could be ambiguous as there are multiple mathematically valid solutions.

arXiv:2502.02613Feb 3, 20250 citations

Quantum Field Theory

Meinard Kuhlmann, Manfred Stöckler

Springer eBooksJan 1, 201814 citations

Conformal Scalar Field Theory from Ising Tricriticality on the Fuzzy Sphere

Joseph C. Taylor, Cristian Voinea, Zlatko Papić +1 more

Free theories are landmarks in the landscape of quantum field theories: their exact solvability serves as a pillar for perturbative constructions of interacting theories. Fuzzy sphere regularization, which combines quantum Hall physics with state-operator correspondence, has recently been proposed as a promising framework for simulating three-dimensional conformal field theories (CFTs), but so far it has not provided access to free theories. We overcome this limitation by designing a bilayer quantum Hall system that hosts an Ising tricritical point-a nontrivial fixed point where first-order and second-order transitions meet-which flows to the conformally coupled scalar theory in the infrared. The critical energy spectrum and operator structure match those at the Gaussian fixed point, providing nonperturbative evidence for the emergence of a free scalar CFT. Our results expand the landscape of CFTs realizable on the fuzzy sphere and demonstrate that even free bosonic theories-previously inaccessible-can emerge from interacting electrons in this framework.Free theories are landmarks in the landscape of quantum field theories: their exact solvability serves as a pillar for perturbative constructions of interacting theories. Fuzzy sphere regularization, which combines quantum Hall physics with state-operator correspondence, has recently been proposed as a promising framework for simulating three-dimensional conformal field theories (CFTs), but so far it has not provided access to free theories. We overcome this limitation by designing a bilayer quantum Hall system that hosts an Ising tricritical point-a nontrivial fixed point where first-order and second-order transitions meet-which flows to the conformally coupled scalar theory in the infrared. The critical energy spectrum and operator structure match those at the Gaussian fixed point, providing nonperturbative evidence for the emergence of a free scalar CFT. Our results expand the landscape of CFTs realizable on the fuzzy sphere and demonstrate that even free bosonic theories-previously inaccessible-can emerge from interacting electrons in this framework.

arXiv:2506.22539Physical Review LettersDec 24, 20257 citations

QUANTUM THEORY OF NON-INTERACTING FIELDS

H. Muirhead

Elsevier eBooksJan 1, 19650 citations

Circuit complexity in quantum field theory

Ro Jefferson, Robert C. Myers

Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.Motivated by recent studies of holographic complexity, we examine the question of circuit complexity in quantum field theory. We provide a quantum circuit model for the preparation of Gaussian states, in particular the ground state, in a free scalar field theory for general dimensions. Applying the geometric approach of Nielsen to this quantum circuit model, the complexity of the state becomes the length of the shortest geodesic in the space of circuits. We compare the complexity of the ground state of the free scalar field to the analogous results from holographic complexity, and find some surprising similarities.

arXiv:1707.08570Journal of High Energy PhysicsOct 1, 2017476 citations

\vec{J}(r,t)

and then dividing by the Born Rule

|\vecΨ(r,t)|^{2}

for velocity arising from Hamilton-Jacobi mechanics. This addition of divergence free probability current would still satisfy the Schrodinger continuity equation. It was found that for a 2D polar system a divergence free probability density is the gradient rotated by the matrix (

S

) applied to a scalar field as such

S\vec{\nabla}φ(r)

. A variety of equations of motion were used such that the dimensions of the equations are equivalent to a velocity, different scalar fields were then derived for a 2D quantum harmonic oscillator. It was found that for each unique velocity there was a unique scalar field associated with it, as such the equation of motion for a particle in De Broglie-Bohm pilot wave theory could be theoretically dependent on any arbitrary number of scalar fields. This ultimately means the equations of motion could be ambiguous as there are multiple mathematically valid solutions.