Pilot wave theory

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Couder experiments, "materializing" the pilot wave model.

In theoretical physics, the pilot wave theory, also known as Bohmian mechanics, was the first known example of a hidden variable theory, presented by Louis de Broglie in 1927. Its more modern version, the de Broglie–Bohm theory, interprets quantum mechanics as a deterministic theory, avoiding troublesome notions such as wave–particle duality, instantaneous wave function collapse, and the paradox of Schrödinger's cat but introducing nonlocality.

The de Broglie–Bohm pilot wave theory is one of several equally valid interpretations of (non-relativistic) quantum mechanics. An extension to the relativistic case has been developed since the 1990s.

History

In his 1926 paper, Max Born suggested that the wave function of Schrödinger's wave equation represents the probability density of finding a particle. From this idea, de Broglie developed the pilot wave theory, and worked out a function for the guiding wave. Initially, de Broglie proposed a double solution approach, in which the quantum object consists of a physical wave (u-wave) in real space which has a spherical singular region that gives rise to particle-like behaviour; in this initial form of his theory he did not have to postulate the existence of a quantum particle. He later formulated it as a theory in which a particle is accompanied by a pilot wave. He presented the pilot wave theory at the 1927 Solvay Conference. However, Wolfgang Pauli raised an objection to it at the conference, saying that it did not deal properly with the case of inelastic scattering. De Broglie was not able to find a response to this objection, and he and Born abandoned the pilot-wave approach. Unlike David Bohm years later, de Broglie did not complete his theory to encompass the many-particle case. The many-particle case shows mathematically that the energy dissipation in inelastic scattering could be distributed to the surrounding field structure by a yet-unknown mechanism of the theory of hidden variables.

In 1932, John von Neumann published a book, part of which claimed to prove that all hidden variable theories were impossible. This result was found to be flawed by Grete Hermann three years later, though this went unnoticed by the physics community for over fifty years
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In 1952, David Bohm, dissatisfied with the prevailing orthodoxy, rediscovered de Broglie's pilot wave theory. Bohm developed pilot wave theory into what is now called the de Broglie–Bohm theory. The de Broglie–Bohm theory itself might have gone unnoticed by most physicists, if it had not been championed by John Bell, who also countered the objections to it. In 1987, John Bell rediscovered Grete Hermann's work, and thus showed the physics community that Pauli's and von Neumann's objections "only" showed that the pilot wave theory did not have locality.

Yves Couder and co-workers in 2010 discovered a macroscopic pilot wave system in the form of walking droplets. This system exhibits behaviour of a pilot wave, heretofore considered to be reserved to microscopic phenomena.

The pilot wave theory

Principles

(a) A walker in a circular corral. Trajectories of increasing length are colour-coded according to the droplet’s local speed (b) The probability distribution of the walker’s position corresponds roughly to the amplitude of the corral’s Faraday wave mode.

The pilot wave theory is a hidden variable theory. Consequently:

• the theory has realism (meaning that its concepts exist independently of the observer);
• the theory has determinism.

The positions and momenta of the particles are considered to be the hidden variables. The observer not only doesn't know the precise value of these variables but more importantly, cannot know them precisely because any measurement disturbs them, this is known as quantum superposition.

A collection of particles has an associated matter wave, which evolves according to the Schrödinger equation. Each particle follows a deterministic trajectory, which is guided by the wave function; collectively, the density of the particles conforms to the magnitude of the wave function. The wave function is not influenced by the particle and can exist also as an empty wave function.

The theory brings to light nonlocality that is implicit in the non-relativistic formulation of quantum mechanics and uses it to satisfy Bell's theorem. These nonlocal effects can be shown to be compatible with the no-communication theorem, which prevents use of them for faster-than-light communication, and so is empirically compatible with relativity.

Mathematical foundations

To derive the de Broglie–Bohm pilot-wave for an electron, the quantum Lagrangian

${\displaystyle L(t)={\frac {1}{2}}mv^{2}-(V+Q),}$

where ${\displaystyle V}$ is the potential energy, ${\displaystyle v}$ is the velocity and ${\displaystyle Q}$ is the potential associated with the quantum force (the particle being pushed by the wave function), is integrated along precisely one path (the one the electron actually follows). This leads to the following formula for the Bohm propagator:

${\displaystyle K^{Q}(X_{1},t_{1};X_{0},t_{0})={\frac {1}{J(t)^{\frac {1}{2}}}}\exp \left[{\frac {i}{\hbar }}\int _{t_{0}}^{t_{1}}L(t)\,dt\right].}$

This propagator allows to track the electron precisely over time under the influence of the quantum potential ${\displaystyle Q}$.

Derivation of the Schrödinger equation

Pilot Wave theory is based on Hamilton–Jacobi dynamics rather than Lagrangian or Hamiltonian dynamics. Using the Hamilton–Jacobi equation

${\displaystyle H\left(\mathbf {q} ,{\partial S \over \partial \mathbf {q} },t\right)+{\partial S \over \partial t}\left(\mathbf {q} ,t\right)=0}$

it is possible to derive the Schrödinger equation:

Consider a classical particle – the position of which is not known with certainty. We must deal with it statistically, so only the probability density ${\displaystyle \rho (x,t)}$ is known. Probability must be conserved, i.e. ${\displaystyle \int \rho \,d^{3}x=1}$ for each ${\displaystyle t}$. Therefore, it must satisfy the continuity equation

${\displaystyle \partial \rho /\partial t=-\nabla \cdot (\rho v)\quad (1)}$

where ${\displaystyle v(x,t)}$ is the velocity of the particle.
In the Hamilton–Jacobi formulation of classical mechanics, velocity is given by ${\displaystyle v(x,t)={\frac {\nabla S(x,t)}{m}}}$ where ${\displaystyle S(x,t)}$ is a solution of the Hamilton-Jacobi equation

${\displaystyle -{\frac {\partial S}{\partial t}}={\frac {\left(\nabla S\right)^{2}}{2m}}+{\tilde {V}}\quad (2)}$

${\displaystyle (1)}$ and ${\displaystyle (2)}$ can be combined into a single complex equation by introducing the complex function ${\displaystyle \psi ={\sqrt {\rho }}e^{\frac {iS}{\hbar }}}$, then the two equations are equivalent to

${\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+{\tilde {V}}-Q\right)\psi \quad }$ with ${\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho }}}{\sqrt {\rho }}}.}$

The time dependent Schrödinger equation is obtained if we start with ${\displaystyle {\tilde {V}}=V+Q}$, the usual potential with an extra quantum potential ${\displaystyle Q}$. The quantum potential is the potential of the quantum force, which is proportional (in approximation) to the curvature of the amplitude of the wave function.

Mathematical formulation for a single particle

The matter wave of de Broglie is described by the time-dependent Schrödinger equation:

${\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V\right)\psi \quad }$

The complex wave function can be represented as:

${\displaystyle \psi ={\sqrt {\rho }}\;\exp \left({\frac {i\,S}{\hbar }}\right)}$

By plugging this into the Schrödinger equation, one can derive two new equations for the real variables. The first is the continuity equation for the probability density${\displaystyle \rho }$: 

${\displaystyle \partial \rho /\partial t+\nabla \cdot (\rho v)=0\;,}$

where the velocity field is defined by the guidance equation

${\displaystyle {\vec {v}}({\vec {r}},t)={\frac {\nabla S({\vec {r}},t)}{m}}\;.}$

According to pilot wave theory, the point particle and the matter wave are both real and distinct physical entities (unlike standard quantum mechanics, where particles and waves are considered to be the same entities, connected by wave–particle duality). The pilot wave guides the motion of the point particles as described by the guidance equation.

Ordinary quantum mechanics and pilot wave theory are based on the same partial differential equation. The main difference is that in ordinary quantum mechanics, the Schrödinger equation is connected to reality by the Born postulate, which states that the probability density of the particle's position is given by ${\displaystyle \rho =|\psi |^{2}}$. Pilot wave theory considers the guidance equation to be the fundamental law, and sees the Born rule as a derived concept.

The second equation is a modified Hamilton–Jacobi equation for the action ${\displaystyle S}$:

${\displaystyle -{\frac {\partial S}{\partial t}}={\frac {\left(\nabla S\right)^{2}}{2m}}+V+Q\;,}$

where Q is the quantum potential defined by

${\displaystyle Q=-{\frac {\hbar ^{2}}{2m}}{\frac {\nabla ^{2}{\sqrt {\rho }}}{\sqrt {\rho }}}}$

By neglecting Q, our equation is reduced to the Hamilton–Jacobi equation of a classical point particle. ( Strictly speaking, this is only a semiclassical limit, because the superposition principle still holds and one needs a decoherence mechanism to get rid of it. Interaction with the environment can provide this mechanism.) So, the quantum potential is responsible for all the mysterious effects of quantum mechanics.

One can also combine the modified Hamilton–Jacobi equation with the guidance equation to derive a quasi-Newtonian equation of motion

${\displaystyle m\,{\frac {d}{dt}}\,{\vec {v}}=-\nabla (V+Q)\;,}$

where the hydrodynamic time derivative is defined as

${\displaystyle {\frac {d}{dt}}={\frac {\partial }{\partial t}}+{\vec {v}}\cdot \nabla \;.}$

Mathematical formulation for multiple particles

The Schrödinger equation for the many-body wave function ${\displaystyle \psi ({\vec {r}}_{1},{\vec {r}}_{2},\cdots ,t)}$ is given by

${\displaystyle i\hbar {\frac {\partial \psi }{\partial t}}=\left(-{\frac {\hbar ^{2}}{2}}\sum _{i=1}^{N}{\frac {\nabla _{i}^{2}}{m_{i}}}+V({\mathbf {r} }_{1},{\mathbf {r} }_{2},\cdots {\mathbf {r} }_{N})\right)\psi }$

The complex wave function can be represented as:

${\displaystyle \psi ={\sqrt {\rho }}\;\exp \left({\frac {i\,S}{\hbar }}\right)}$

The pilot wave guides the motion of the particles. The guidance equation for the jth particle is:

${\displaystyle {\vec {v}}_{j}={\frac {\nabla _{j}S}{m_{j}}}\;.}$

The velocity of the jth particle explicitly depends on the positions of the other particles. This means that the theory is nonlocal.

Empty wave function

Lucien Hardy and John Stewart Bell have emphasized that in the de Broglie–Bohm picture of quantum mechanics there can exist empty waves, represented by wave functions propagating in space and time but not carrying energy or momentum, and not associated with a particle. The same concept was called ghost waves (or "Gespensterfelder", ghost fields) by Albert Einstein. The empty wave function notion has been discussed controversially. In contrast, the many-worlds interpretation of quantum mechanics does not call for empty wave functions.

Holomorphic function

From Wikipedia, the free encyclopedia

 

A rectangular grid (top) and its image under a Conformal map f (bottom).

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative in a neighborhood is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series (analytic). Holomorphic functions are the central objects of study in complex analysis.

Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighborhood of each point in its domain. The fact that all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.

Holomorphic functions are also sometimes referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point z₀" means not just differentiable at z₀, but differentiable everywhere within some neighborhood of z₀ in the complex plane.

Definition

The function ${\displaystyle f(z)={\bar {z}}}$ is not complex-differentiable at zero, because as shown above, the value of ${\displaystyle f(z)-f(0) \over z-0}$ varies depending on the direction from which zero is approached. Along the real axis, f equals the function g(z) = z and the limit is 1, while along the imaginary axis, f equals h(z) = −z and the limit is −1. Other directions yield yet other limits.

Given a complex-valued function f of a single complex variable, the derivative of f at a point z₀ in its domain is defined by the limit

${\displaystyle f'(z_{0})=\lim _{z\to z_{0}}{f(z)-f(z_{0}) \over z-z_{0}}.}$

This is the same as the definition of the derivative for real functions, except that all of the quantities are complex. In particular, the limit is taken as the complex number z approaches z₀, and must have the same value for any sequence of complex values for z that approach z₀ on the complex plane. If the limit exists, we say that f is complex-differentiable at the point z₀. This concept of complex differentiability shares several properties with real differentiability: it is linear and obeys the product rule, quotient rule, and chain rule.

If f is complex differentiable at every point z₀ in an open set U, we say that f is holomorphic on U. We say that f is holomorphic at the point z₀ if it is holomorphic on some neighborhood of z₀. We say that f is holomorphic on some non-open set A if it is holomorphic in an open set containing A. As a pathological non-example, the function given by f(z) = |z|² is complex differentiable at exactly one point (z₀ = 0), and for this reason, it is not holomorphic at 0 because there is no open set around 0 on which f is complex differentiable.

The relationship between real differentiability and complex differentiability is the following. If a complex function f(x + i y) = u(x, y) + i v(x, y) is holomorphic, then u and v have first partial derivatives with respect to x and y, and satisfy the Cauchy–Riemann equations:

${\displaystyle {\frac {\partial u}{\partial x}}={\frac {\partial v}{\partial y}}\qquad {\mbox{and}}\qquad {\frac {\partial u}{\partial y}}=-{\frac {\partial v}{\partial x}}\,}$

or, equivalently, the Wirtinger derivative of f with respect to the complex conjugate of z is zero:

${\displaystyle {\frac {\partial f}{\partial {\overline {z}}}}=0,}$

which is to say that, roughly, f is functionally independent from the complex conjugate of z.
If continuity is not given, the converse is not necessarily true. A simple converse is that if u and v have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then f is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if f is continuous, u and v have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then f is holomorphic.

Terminology

The word "holomorphic" was introduced by two of Cauchy's students, Briot (1817–1882) and Bouquet (1819–1895), and derives from the Greek ὅλος (holos) meaning "entire", and μορφή (morphē) meaning "form" or "appearance".

Today, the term "holomorphic function" is sometimes preferred to "analytic function", as the latter is a more general concept. This is also because an important result in complex analysis is that every holomorphic function is complex analytic, a fact that does not follow obviously from the definitions. The term "analytic" is however also in wide use.

Properties

Because complex differentiation is linear and obeys the product, quotient, and chain rules; the sums, products and compositions of holomorphic functions are holomorphic, and the quotient of two holomorphic functions is holomorphic wherever the denominator is not zero.

If one identifies C with R², then the holomorphic functions coincide with those functions of two real variables with continuous first derivatives which solve the Cauchy–Riemann equations, a set of two partial differential equations.

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R². In other words, if we express a holomorphic function f(z) as u(x, y) + i v(x, y) both u and v are harmonic functions, where v is the harmonic conjugate of u.
Cauchy's integral theorem implies that the contour integral of every holomorphic function along a loop vanishes:

${\displaystyle \oint _{\gamma }f(z)\,dz=0.}$

Here γ is a rectifiable path in a simply connected open subset U of the complex plane C whose start point is equal to its end point, and f : U → C is a holomorphic function.

Cauchy's integral formula states that every function holomorphic inside a disk is completely determined by its values on the disk's boundary. Furthermore: Suppose U is an open subset of C, f : U → C is a holomorphic function and the closed disk D = {z : |z − z₀| ≤ r} is completely contained in U. Let γ be the circle forming the boundary of D. Then for every a in the interior of D:

${\displaystyle f(a)={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{z-a}}\,dz}$

where the contour integral is taken counter-clockwise.

The derivative f′(a) can be written as a contour integral using Cauchy's differentiation formula:

${\displaystyle f'(a)={1 \over 2\pi i}\oint _{\gamma }{f(z) \over (z-a)^{2}}\,dz,}$

for any simple loop positively winding once around a, and

${\displaystyle f'(a)=\lim \limits _{\gamma \to a}{\frac {i}{2{\mathcal {A}}(\gamma )}}\oint _{\gamma }f(z)d{\bar {z}},}$

for infinitesimal positive loops γ around a.

In regions where the first derivative is not zero, holomorphic functions are conformal in the sense that they preserve angles and the shape (but not size) of small figures.

Every holomorphic function is analytic. That is, a holomorphic function f has derivatives of every order at each point a in its domain, and it coincides with its own Taylor series at a in a neighborhood of a. In fact, f coincides with its Taylor series at a in any disk centered at that point and lying within the domain of the function.

From an algebraic point of view, the set of holomorphic functions on an open set is a commutative ring and a complex vector space. Additionally, the set of holomorphic functions in an open set U is an integral domain if and only if the open set U is connected.  In fact, it is a locally convex topological vector space, with the seminorms being the suprema on compact subsets.

From a geometric perspective, a function f is holomorphic at z₀ if and only if its exterior derivative df in a neighborhood U of z₀ is equal to f′(z) dz for some continuous function f′. It follows from

${\displaystyle \textstyle 0=d^{2}f=d(f^{\prime }dz)=df^{\prime }\wedge dz}$

that df′ is also proportional to dz, implying that the derivative f′ is itself holomorphic and thus that f is infinitely differentiable. Similarly, the fact that d(f dz) = f′ dz ∧ dz = 0 implies that any function f that is holomorphic on the simply connected region U is also integrable on U. (For a path γ from z₀ to z lying entirely in U, define

${\displaystyle \textstyle F_{\gamma }(z)=F_{0}+\int _{\gamma }fdz}$;

in light of the Jordan curve theorem and the generalized Stokes' theorem, F_γ(z) is independent of the particular choice of path γ, and thus F(z) is a well-defined function on U having F(z₀) = F₀ and dF = f dz.)

Examples

All polynomial functions in z with complex coefficients are holomorphic on C, and so are sine, cosine and the exponential function. (The trigonometric functions are in fact closely related to and can be defined via the exponential function using Euler's formula). The principal branch of the complex logarithm function is holomorphic on the set C ∖ {z ∈ R : z ≤ 0}. The square root function can be defined as

${\displaystyle {\sqrt {z}}=e^{{\frac {1}{2}}\log z}}$

and is therefore holomorphic wherever the logarithm log(z) is. The function 1/z is holomorphic on {z : z ≠ 0}.

As a consequence of the Cauchy–Riemann equations, a real-valued holomorphic function must be constant. Therefore, the absolute value of z, the argument of z, the real part of z and the imaginary part of z are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate z formed by complex conjugation.

Several variables

The definition of a holomorphic function generalizes to several complex variables in a straightforward way. Let D denote an open subset of Cⁿ, and let f : D → C. The function f is analytic at a point p in D if there exists an open neighborhood of p in which f is equal to a convergent power series in n complex variables. Define f to be holomorphic if it is analytic at each point in its domain. Osgood's lemma shows (using the multivariate Cauchy integral formula) that, for a continuous function f, this is equivalent to f being holomorphic in each variable separately (meaning that if any n − 1 coordinates are fixed, then the restriction of f is a holomorphic function of the remaining coordinate). The much deeper Hartogs' theorem proves that the continuity hypothesis is unnecessary: f is holomorphic if and only if it is holomorphic in each variable separately.

More generally, a function of several complex variables that is square integrable over every compact subset of its domain is analytic if and only if it satisfies the Cauchy–Riemann equations in the sense of distributions.

Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are Reinhardt domains, the simplest example of which is a polydisk. However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such a set is called a domain of holomorphy.

Extension to functional analysis

The concept of a holomorphic function can be extended to the infinite-dimensional spaces of functional analysis. For instance, the Fréchet or Gâteaux derivative can be used to define a notion of a holomorphic function on a Banach space over the field of complex numbers.