Spinors ψj of SU(2) represent group elements, i.e. three-dimensional rotations Rj , because they are shorthands of SU(2) representation matrices Rj obtained by taking their first columns. We explain that making linear combinations of spinors of SU(2) is feasible algebraically but geometrically meaningless. E.g. ψ1 + ψ2 ∈ C 2 would have to correspond to R1 + R2. But in SO(3), R1 + R2 would be a function that transforms r ∈ R 3 into R1(r) + R2(r), while in SU(2) it would correspond to a different function that is up to a multiplication constant equal to S : r → S(r) = R1(r) + R2(r) + 2 [ cos(ϕ1/2) cos(ϕ2/2) − s1•s2 ] r + 2 sin(ϕ1/2) sin(ϕ2/2)[ (s2•r) s1 + (s1•r) s2 ] + 2 sin(ϕ1/2) cos(ϕ2/2)(s1 ∧ r) + 2 cos(ϕ1/2) sin(ϕ2/2)(s2 ∧ r). The multiplication constant must be chosen such that |r| is preserved. Such extensions by linear combinations for spinors from a manifold to an embedding vector space are therefore mindless algebra with a spurious, conceptually impenetrable geometrical counterpart. This should not surprise anybody because the group axioms only define products of group elements, not linear combinations of them. Redefining spinors as vectors of a Hilbert space is therefore a purely formal, would-be scholar generalization.
Spinors ψj of SU(2) represent group elements, i.e. three-dimensional rotations Rj , because they are shorthands of SU(2) representation matrices Rj obtained by taking their first columns. We explain that making linear combinations of spinors of SU(2) is feasible algebraically but geometrically meaningless. E.g. ψ1 + ψ2 ∈ C 2 would have to correspond to R1 + R2. But in SO(3), R1 + R2 would be a function that transforms r ∈ R 3 into R1(r) + R2(r), while in SU(2) it would correspond to a different function that is up to a multiplication constant equal to S : r → S(r) = R1(r) + R2(r) + 2 [ cos(ϕ1/2) cos(ϕ2/2) − s1•s2 ] r + 2 sin(ϕ1/2) sin(ϕ2/2)[ (s2•r) s1 + (s1•r) s2 ] + 2 sin(ϕ1/2) cos(ϕ2/2)(s1 ∧ r) + 2 cos(ϕ1/2) sin(ϕ2/2)(s2 ∧ r). The multiplication constant must be chosen such that |r| is preserved. Such extensions by linear combinations for spinors from a manifold to an embedding vector space are therefore mindless algebra with a spurious, conceptually impenetrable geometrical counterpart. This should not surprise anybody because the group axioms only define products of group elements, not linear combinations of them. Redefining spinors as vectors of a Hilbert space is therefore a purely formal, would-be scholar generalization.
This paper aims at explaining that the key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R 4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R 3. In the three-dimensional rotation group the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding acquired to derive the free-space Dirac equation from scratch proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine's statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.
Spin is what makes the world go 'round. The Groups SO (3) and SU (2) How does SU(2) act on its Lie algebra? For physical and mathematical motivation for this section (which involves nonrelativistic quantum mechanics) we refer the reader to some remarks of Feynman and of Weyl. Specifically, Feynman, in his section entitled “Degeneracy,” shows that a process involving a specific choice of direction in space requires that the process be described not by a single wave function ψ but rather by a multicomponent column vector of wave functions Ψ = (ψ 1 , …, ψ N ) T . He then indicates, roughly speaking, that since the physics cannot depend on the choice of cartesian coordinates ( x 1 , x 2 , x 3 ) of space, the N -tuples must transform under some representation ρ: SO (3) → U ( N ) of the rotation group SO (3) of space. This is not quite accurate; since e i γ Ψ represents the same wave function (when γ is a constant), ρ is only a “ray” representation, ρ( g )ρ( h ) = e i γ( g,h ) ρ( gh ) for a function γ( g, h ). Weyl shows that this can be made into a genuine representation, except that it is (perhaps) double-valued. We shall show in this section that there is a natural 2 : 1 homomorphism π of the special unitary group SU (2) onto SO (3), thus yielding a (perhaps double valued) representation of SU (2) into U ( N ).
This paper aims at explaining that a key to understanding quantum mechanics (QM) is a perfect geometrical understanding of the spinor algebra that is used in its formulation. Spinors occur naturally in the representation theory of certain symmetry groups. The spinors that are relevant for QM are those of the homogeneous Lorentz group SO(3,1) in Minkowski space-time R4 and its subgroup SO(3) of the rotations of three-dimensional Euclidean space R3. In the three-dimensional rotation group, the spinors occur within its representation SU(2). We will provide the reader with a perfect intuitive insight about what is going on behind the scenes of the spinor algebra. We will then use the understanding that is acquired to derive the free-space Dirac equation from scratch, proving that it is a description of a statistical ensemble of spinning electrons in uniform motion, completely in the spirit of Ballentine’s statistical interpretation of QM. This is a mathematically rigorous proof. Developing this further, we allow for the presence of an electromagnetic field. We can consider the result as a reconstruction of QM based on the geometrical understanding of the spinor algebra. By discussing a number of problems in the interpretation of the conventional approach, we illustrate how this new approach leads to a better understanding of QM.
Spinors are used for the computation of probability in quantum mechanics. They are treated as elements of a complex vector space in contrast to a real vector space. Spinor theory is abstract mathematics with an ambiguous interpretation. The overall phase of spinors does not affect the computation of probability in quantum mechanics. If two spinors have an overall phase of imaginary number i, then they can be treated like vectors in a real vector space. The square of the magnitude of the number, which is the sum of dot product and the term having basis vector i of cross product of two such vectors, is equal to the probability computed using the corresponding spinors. Therefore, the geometry of such spinors can be easily depicted in a three-dimensional space like vectors in a real vector space. Spinors are not isotropic vectors in Hilbert space. The sum of dot product and cross product of two complex numbers is equivalent to the quotient of the division of one complex number by another. Similarly, the sum of dot product and cross product of two vectors is the quotient of the division of one vector by another. Using the rules of division of vectors, we can find the rules of multiplication of vectors. The rules of multiplication and division of basis vector i match the rules of multiplication and division of imaginary number i. Therefore, basis vector i is nothing but imaginary number i. Multiplication of dot and cross product of two vectors to the second vector will give us the components of the first vector that are parallel and orthogonal to the second vector. Vectors are also made up of complex numbers like spinors. The reason for finding dot product of complex numbers in the process of computing the probability is to ignore the overall phase in contrast to the phase difference. This is misconstrued as complex vector space in quantum mechanics. 3-D number which is nothing but a spinor with new notation is an extension of vector algebra. It can have a real number as a term in addition to the terms i, j and k. In all other respects, it is a vector. 3-D numbers are part of the number system like real numbers and complex numbers. The real numbers are one dimensional numbers, the complex numbers are two-dimensional numbers and the 3-D numbers as well as vectors and spinors are three-dimensional numbers. As polarisation and spin are one and the same, we can straight away apply 3-D numbers to polarisation of light. 3-D numbers will greatly simplify the study of areas of science where three-dimensional space is involved.
Spinors are used for the computation of probability in quantum mechanics. They are treated as elements of a complex vector space in contrast to a real vector space. Spinor theory is abstract mathematics with an ambiguous interpretation. The overall phase of spinors does not affect the computation of probability in quantum mechanics. If two spinors have an overall phase of imaginary number , then they can be treated like vectors in a real vector space. The square of the magnitude of the number, which is the sum of dot product and the term having basis vector of cross product of two such vectors, is equal to the probability computed using the corresponding spinors. Therefore, the geometry of such spinors can be easily depicted in a three-dimensional space like vectors in a real vector space. Spinors are not isotropic vectors in Hilbert space. The sum of dot product and cross product of two complex numbers is equivalent to the quotient of the division of one complex number by another. Similarly, the sum of dot product and cross product of two vectors is the quotient of the division of one vector by another. Using the rules of division of vectors, we can find the rules of multiplication of vectors. The rules of multiplication and division of basis vector match the rules of multiplication and division of imaginary number . Therefore, basis vector is nothing but imaginary number . Multiplication of dot and cross product of two vectors to the second vector will give us the components of the first vector that are parallel and orthogonal to the second vector. Vectors are also made up of complex numbers like spinors. The reason for finding dot product of complex numbers in the process of computing the probability is to ignore the overall phase in contrast to the phase difference. This is misconstrued as complex vector space in quantum mechanics. 3-D number which is nothing but a spinor with new notation is an extension of vector algebra. It can have a real number as a term in addition to the terms , and . In all other respects, it is a vector. 3-D numbers are part of the number system like real numbers and complex numbers. The real numbers are one dimensional numbers, the complex numbers are two-dimensional numbers and the 3-D numbers as well as vectors and spinors are three-dimensional numbers. As polarisation and spin are one and the same, we can straight away apply 3-D numbers to polarisation of light. 3-D numbers will greatly simplify the study of areas of science where three-dimensional space is involved.
Hilbert space and inner product In Ch. 2 it was noted that quantum wave functions form a linear space in the sense that multiplying a function by a complex number or adding two wave functions together produces another wave function. It was also pointed out that a particular quantum state can be represented either by a wave function ψ( x ) which depends upon the position variable x , or by an alternative function ψ( p ) of the momentum variable p . It is convenient to employ the Dirac symbol |ψ〉, known as a “ket”, to denote a quantum state without referring to the particular function used to represent it. The kets, which we shall also refer to as vectors to distinguish them from scalars , which are complex numbers, are the elements of the quantum Hilbert space H . (The real numbers form a subset of the complex numbers, so that when a scalar is referred to as a “complex number”, this includes the possibility that it might be a real number.) If α is any scalar (complex number), the ket corresponding to the wave function ɑψ( x ) is denoted by α|ψ〉, or sometimes by |ψ〉α, and the ket corresponding to ø( x )+ψ( x ) is denoted by |ø〉+|ψ〉 or |ψ〉+|ø〉, and so forth. This correspondence could equally well be expressed using momentum wave functions, because the Fourier transform, (2.15) or (2.16), is a linear relationship between ψ( x ) and ( p ), so that αø( x )+βψ( x ) and αϕ( p ) + βψ( p ) correspond to the same quantum state α|ψ〉+β|ø〉.