Theory of Stochastic Schrödinger Equation in Complex Vector Space

Abstract

A generalized Schrödinger equation containing correction terms to classical kinetic energy, has been derived in the complex vector space by considering an extended particle structure in stochastic electrodynamics with spin. The correction terms are obtained by considering the internal complex structure of the particle which is a consequence of stochastic average of particle oscillations in the zeropoint field. Hence, the generalised Schrödinger equation may be called stochastic Schrödinger equation. It is found that the second order correction terms are similar to corresponding relativistic corrections. When higher order correction terms are neglected, the stochastic Schrödinger equation reduces to normal Schrödinger equation. It is found that the Schrödinger equation contains an internal structure in disguise and that can be revealed in the form of internal kinetic energy. The internal kinetic energy is found to be equal to the quantum potential obtained in the Madelung fluid theory or Bohm statistical theory. In the rest frame of the particle, the stochastic Schrödinger equation reduces to a Dirac type equation and its Lorentz boost gives the Dirac equation. Finally, the relativistic Klein–Gordon equation is derived by squaring the stochastic Schrödinger equation. The theory elucidates a logical understanding of classical approach to quantum mechanical foundations.

Appendix: Algebra of Complex Vectors

The geometric algebra or Clifford algebra is found to be a superior algebra than the vector algebra and it is being used by a growing number of mathematicians and physicists today [54, 55]. A detailed account of geometric algebra and its applications to physics is given in the book by Doran and Lasenby [56]. An introduction to geometric algebra and the algebra of complex vectors are considered in this appendix.

The geometric product of two vectors $\mathbf {a}$ and $\mathbf {b}$ is defined as

$$\begin{aligned} \mathbf {a}\mathbf {b}=\mathbf {a}{\cdot }\mathbf {b}+\mathbf {a}\wedge \mathbf {b}, \end{aligned}$$

(68)

where the scalar product or symmetric product is defined as

$$\begin{aligned} \mathbf {a}{\cdot }\mathbf {b}= \frac{1}{2}(\mathbf {a}\mathbf {b}+\mathbf {b}\mathbf {a}), \end{aligned}$$

(69)

and the wedge product or outer product is defined as

$$\begin{aligned} \mathbf {a} \wedge \mathbf {b}= \frac{1}{2}(\mathbf {a}\mathbf {b}-\mathbf {b}\mathbf {a}). \end{aligned}$$

(70)

Changing the order of vectors is called reversion operation and it is denoted by an over bar.

$$\begin{aligned} \overline{\mathbf {a}\mathbf {b}}=\mathbf {b}\mathbf {a} =\mathbf {a}{\cdot }\mathbf {b}-\mathbf {a}\wedge \mathbf {b}, \end{aligned}$$

(71)

A set of unit right handed basis vectors $\lbrace \mathbf {\sigma }_k;k=1,2,3\rbrace$ is considered to span the three dimensional space. A pseudoscalar in three dimensional space is defined as $\mathbf {i}=\mathbf {\sigma }_1\mathbf {\sigma }_2\mathbf {\sigma }_3$ and it represents a unit oriented volume. Multiplying vectors $\mathbf {\sigma }_k$ by pseudoscalar form unit bivectors, $B_k=\mathbf {i}\mathbf {\sigma }_k=\mathbf {\sigma }_i\mathbf {\sigma }_j.$ Each unit bivector represents an oriented plane. The set of elements $\lbrace 1,\mathbf {\sigma }_k,B_k,\mathbf {i};k=1,2,3\rbrace$ form geometric algebra of Euclidean space. A general element in geometric algebra is called a multivector and it is a sum of a scalar, vector, bivector and trivector.

$$\begin{aligned} M=\alpha +\mathbf {a}+ \mathbf {ib} + \mathbf {i} \delta , \end{aligned}$$

(72)

where $\alpha$ and $\delta$ are scalars, $\mathbf {a}$ and $\mathbf {b}$ are vectors and $\mathbf {i}\delta$ is a trivector. A complex vector is defined as a sum of a vector and a bivector.

$$\begin{aligned} Z=\mathbf {a}+\mathbf {ib} \end{aligned}$$

(73)

The advantage of this definition of complex vector gives an additional geometric understanding of orientation and rotation in space. A reversion operation on a complex vector changes the sign of bivector.

$$\begin{aligned} \bar{Z}=\mathbf {a}-\mathbf {ib} \end{aligned}$$

(74)

The complex vector $\bar{Z}$ is known as complex conjugate of Z. Two complex vectors are equal when their vector and bivector parts are equal. The inner product of two complex vectors $Z=\mathbf {a}+\mathbf {ib}$ and $Y=\mathbf {c}+\mathbf {id}$ can be expressed as

$$\begin{aligned} Z.Y=(\mathbf {a}{\cdot }\mathbf {c}-\mathbf {b}{\cdot }\mathbf {d})+\mathbf {i}(\mathbf {b}{\cdot }\mathbf {c}+\mathbf {a}{\cdot }\mathbf {d})= \alpha + \mathbf {i}\beta . \end{aligned}$$

(75)

Thus the scalar product of two complex vectors is a complex scalar. The outer product of complex vectors Z and Y is

$$\begin{aligned} Z\wedge Y=(\mathbf {a}\wedge \mathbf {c}-\mathbf {b}\wedge \mathbf {d})+\mathbf {i}(\mathbf {b}\wedge \mathbf {c}+\mathbf {a}\wedge \mathbf {d}) . \end{aligned}$$

(76)

The term $(\mathbf {a}\wedge \mathbf {c}-\mathbf {b}\wedge \mathbf {d})$ is a bivector and the term $\mathbf {i}(\mathbf {b}\wedge \mathbf {c}+\mathbf {a}\wedge \mathbf {d})$ is vector. Thus, the outer product of two complex vectors is a complex vector. From the above two products one can see that the geometric product of two complex vectors is a combination of a scalar, vector, bivector and trivector parts. Thus the geometric product of two complex vectors is a multivector. The inner and outer products of complex vectors are in general known as symmetric and asymmetric products respectively. Two complex vectors $Z=\mathbf {a}+\mathbf {ib}$ and $Y=\mathbf {c}+\mathbf {id}$ are said to be perpendicular when the condition $\mathbf {a}{\cdot } \mathbf {c}=0$ is satisfied and they are parallel when the condition $\mathbf {a} \wedge \mathbf {c}=0$ is satisfied. The square of a complex vector is a complex scalar.

$$\begin{aligned} Z^2=(\mathbf {a}+\mathbf {ib})(\mathbf {a}+\mathbf {ib})=a^2 +b^2 +2\mathbf {i}(\mathbf {a}{\cdot }\mathbf {b}) \end{aligned}$$

(77)

Consider that the vectors $\mathbf {a}$ and $\mathbf {b}$ are orthogonal to each other. Then the scalar product $\mathbf {a}{\cdot }\mathbf {b}=0$. In this case, the complex vector represents an oriented directional ellipse. The bivector $\mathbf {i b}$ represents an oriented plane and the vector $\mathbf {a}$ lies in the plane of $\mathbf {i b}$. The rotation in the plane $\mathbf {i b}$ is counterclockwise for the complex vector Z and clockwise for $\bar{Z}$. The product of a complex vector with its conjugate contains scalar and vector parts. The products $Z\bar{Z}$ and $\bar{Z}Z$ are written in the following form.

$$\begin{aligned} \bar{Z}Z= a^2 +b^2 +2\mathbf {i}(\mathbf {a}\wedge \mathbf {b}) \end{aligned}$$

(78)

$$\begin{aligned} Z\bar{Z}= a^2 +b^2 -2\mathbf {i}(\mathbf {a}\wedge \mathbf {b}) \end{aligned}$$

(79)

Since, $\mathbf {i}$ is a pseudoscalar which commutes with all vectors in three dimensional space, the quantity $2\mathbf {i}(\mathbf {a}\wedge \mathbf {b})$ is a vector and it is normal to the orientation of the bivector $\mathbf {a}\wedge \mathbf {b}$. The scalar part of (78) is equal to the scalar product of $\bar{Z}$ and Z.

$$\begin{aligned} \bar{Z}.Z= \frac{1}{2} (\bar{Z}Z+Z\bar{Z})= a^2 +b^2 \end{aligned}$$

(80)

The vector part of (78) is equal to the outer product of $\bar{Z}$ and Z.

$$\begin{aligned} \bar{Z} \wedge Z= \frac{1}{2} (\bar{Z}Z-Z\bar{Z})= 2\mathbf {i}(\mathbf {a}\wedge \mathbf {b}) \end{aligned}$$

(81)

In the case when the magnitudes of vectors $\mathbf {a}$ and $\mathbf {b}$ are equal, the complex vector represents an oriented directional circle. Then the square of complex vector $Z^2 = \bar{Z}^2 = 0$ and therefore in this case the complex vector may be called complex null vector.

When the third direction is chosen normal to the bivector plane $\mathbf {a}\wedge \mathbf {b}$, the complex vector Z and its conjugate $\bar{Z}$ represent a physical space. If we choose a set of orthonormal right handed unit vectors $\lbrace \mathbf {\sigma }_k;k=1,2,3\rbrace$ along the direction of vectors $\mathbf {a}$, $\mathbf {b}$ and $2\mathbf {i}(\mathbf {a}\wedge \mathbf {b})$, the unit vectors $\mathbf {\sigma }_k$ can be expressed in terms of complex vectors Z and $\bar{Z}$.

$$\begin{aligned} \mathbf {\sigma }_1 = \frac{Z+\bar{Z}}{2a}; \mathbf {\sigma }_2 =\frac{Z-\bar{Z}}{2a}; \mathbf {\sigma }_3 =\frac{Z\bar{Z}-\bar{Z}Z}{4ab}; 1 =\frac{Z\bar{Z}+\bar{Z}Z}{4ab} \end{aligned}$$

(82)

The basis elements $\lbrace 1, \mathbf {\sigma }_k \rbrace$ form a closed complex four dimensional linear space. A complete version of complex vector algebra is elaborated in [44].