A Complete Inner Product Space with Dirac's Bracket Notation

A mathematical connection between the bracket notation of quantum mechanics and quaternions is detailed. It will be argued that quaternions have the properties of a complete inner-product space (a Banach space for the field of quaternions). A central issue is the definition of the square of the norm. In quantum mechanics:

the norm of phi squared = <phi | phi>

In this notebook, the following assertion will be examined (star is the conjugate, so the vector flips signs):

the norm of \(t, X\) squared = \(t, X\) conjugated times \(t, X\) times \(t, X\) conjugated times \(t, X\)

The inner-product of two quaternions is defined here as the transpose (or conjugate) of the first quaternion multiplied by the second. The inner product of a function with itself is the norm.

The Positive Definite Norm of a Quaternion

The square of the norm of a quaternion can only be zero if every element is zero, otherwise it must have a positive value.

\(t, X\) conjugated times \(t, X\) = \(t squared +X dot X, 0\)

This is the standard Euclidean norm for a real 4-dimensional vector space.

The Euclidean inner-product of two quaternions can take on any value, as is the case in quantum mechanics for <phi|theta>. The adjective "Euclidean" is used to distinguish this product from the Grassman inner- product which plays a central role in special relativity (see alternative algebra for boosts).

Completeness

With the topology of a Euclidean norm for a real 4-dimensional vector space, quaternions are complete.

Quaternions are complete in a manner required to form a Banach space if there exists a neighborhood of any quaternion x such that there is a set of quaternions y

the norm of x - y squared is less than epsilon to the fourth

for some fixed value of epsilon.

Construct such a neighborhood.

\(\(t, X\) - epsilon \(t, X\) over 4)conjugated times \(\(t, X\) - epsilon \(t, X\) over 4) times \(\(t, X\) - epsilon \(t, X\) over 4)conjugated times \(\(t, X\) - epsilon \(t, X\) over 4) =

= \(epsilon to the fourth over 4, 0, 0, 0\) is less than \(epsilon to the fourth, 0, 0, 0\)

An infinite number of quaternions exist in the neighborhood.

Any polynomial equation with quaternion coefficients has a quaternion solution in x (a proof done by Eilenberg and Niven in 1944, cited in Birkhoff and Mac Lane's "A Survey of Modern Algebra.")

Identities and Inequalities

The following identities and inequalities emanate from the properties of a Euclidean norm. They are worked out for quaternions here in detail to solidify the connection between the machinery of quantum mechanics and quaternions.

The conjugate of the square of the norm equals the square of the norm of the two terms reversed.

<phi | phi> conjugated = <phi | phi>

For quaternions,

\(\(t, X\) conjugated times \(t prime, X prime\))conjugated = \(t t prime+ X dot X prime, -t X prime + t prime X + X cross X prime\)

\(t prime, X prime\) conjugated times \(t, X\) = \(t prime t + X prime dot X, t prime X - X prime t - X prime Cross X\)

These are identical, because the terms involving the cross produce will flip signs when their order changes.

For products of squares of norms in quantum mechanics,

<psi phi | psi phi> = <psi | phi> times <psi | phi>

This is also the case for quaternions.

<\(t, X\)\(t prime, X prime\) | \(t, X\)\(t prime, X prime\)> =

= \(\(t, X\) times \(t prime, X prime\)) conjugated times \(t, X\) times \(t prime, X prime\) =

= \(t prime, X prime\) conjugated times \(t, X\) conjugated \(t, X\) times \(t prime, X prime\) =

= \(t prime, X prime\) conjugated times \(t squared +x squared +y squared +z squared, 0, 0, 0\) times \(t prime, X prime\) =

= \(t squared +x squared +y squared +z squared, 0, 0, 0\) times \(t prime, X prime\) conjugated times \(t prime, X prime\) =

= \(t, X\) conjugated times \(t, X\) times \(t prime, X prime\) conjugated times \(t prime, X prime\) =

=<\(t, X\) | \(t, X\)> times <\(t prime, X prime\) | \(t prime, X prime\)>

The triangle inequality in quantum mechanics:

<psi + phi | psi + phi> squared is less than \(<psi | psi>+ <phi | phi>\) squared

For quaternions,

<\(t, X\)+\(t prime, X prime\) | \(t, X\)+\(t prime, X prime\)> squared =

= \(\(t+t prime, X+X prime\) conjugated times\(t+t prime, X+X prime\)) squared

= \(t squared + t prime squared + X squared + X prime squared + 2 t t prime + 2X dot X, 0\) squared

is less than

= \(t squared + t prime squared + X squared + X prime squared + 2 times the square root of the norm of \(t, X\) times \(t prime, X prime\), 0) squared =

\(<\(t, X\) | \(t, X\)> + <\(t prime, X prime\) | \(t prime, X prime\)>) squared

If the signs of each pair of component are the same, the two sides will be equal. If the signs are different (a t and a -t for example), then the cross terms will cancel on the left hand side of the inequality, making it smaller than the right hand side where terms never cancel because there are only squared terms.

The Schwarz inequality in quantum mechanics is analogous to dot products and cosines in Euclidean space.

the absolute value of <psi | phi> squared is less than or equal to <psi | psi> times <phi | phi>

Let a third wave function, chi, be the sum of these two with an arbitrary parameter lambda.

chi is defined to be psi + lambda phi

The norm of chi will necessarily be greater than zero.

psi + lambda phi conjugated times psi + lambda phi = psi squared + lambda psi conjugated time phi + lambda conjugated times phi conjugated times psi + lambda conjugated times lambda times phi conjugated times phi is greater than or equal to zero

Choose the value for lambda that helps combine all the terms containing lambda.

lambda goes to - phi conjugated time psi over phi conjugated times phi

psi conjugated time psi - phi conjugated times psi times psi conjugated times phi over phi conjugated times phi

Multiply through by the denominator, separate the two resulting terms and do some minor rearranging.

\(psi conjugated phi\) conjugated times psi conjugated phi is less than or equal to psi conjugated psi times phi conjugated phi

This is now the Schwarz inequality.

Another inequality:

2 Re <phi | phi> <= <phi | phi> + <phi | phi>

Examine the square of the norm of the difference between two quaternions which is necessarily equal to or greater than zero.

0 <= <\(t, X\) - \(t prime, X prime\) | \(t, X\)-\(t prime, X prime\)>

= \(\(t - t prime\) squared + \(X-X prime\).\(X-X prime\), 0)

The cross terms can be put on the other side of inequality, changing the sign, and leaving the sum of two norms behind.

\(2 \(t t prime + X dot X prime\), 0) <= \(t squared + x squared + t prime squared + X prime squared , 0\)

2 Re <\(t, X\) | \(t prime, X prime\)> <= <\(t, X\) | \(t, X\)> + <\(t prime, X prime\) | \(t prime, X prime\)>

The inequality holds.

The parallelogram law:

<phi +phi | phi +phi> + <phi -phi | phi -phi> = 2 <phi | phi> + 2 <phi | phi>

Test the quaternion norm

<\(t, X\) + \(t prime, X prime\) | \(t, X\)+ \(t prime, X prime\)> + <\(t, X\) - \(t prime, X prime\) | \(t, X\) - \(t prime, X prime\)> =

= \(\(t + t prime\) squared + \(X+X prime\).\(X+X prime\), 0) + \(\(t - t prime\) squared + \(X-X prime\).\(X-X prime\), 0) =

= 2 \(t squared + X squared + t prime squared + X prime squared , 0\) =

= 2 <\(t, X\) | \(t, X\)> + 2 <\(t prime, X prime\) | \(t prime, X prime\)>

This is twice the square of the norms of the two separate components.

Implications

In the case for special relativity, it was noticed that by simply squaring a quaternion, the resulting first term was the Lorentz invariant interval. From that solitary observation, the power of a mathematical field was harnessed to solve a wide range of problems in special relativity.

In a similar fashion, it is hoped that because the product of a transpose of a quaternion with a quaternion has the properties of a complete inner product space, the power of the mathematical field of quaternions can be used to solve a wide range of problems in quantum mechanics. This is an important area for further research.

Note: this goal is different from the one Stephen Adler sets out in "Quaternionic Quantum Mechanics and Quantum Fields." He tries to substitute quaternions in the place of complex numbers in the standard Hilbert space formulation of quantum mechanics. The analytical properties of quaternions do not play a critical role. It is the properties of the Hilbert space over the field of quaternions that is harnessed to solve problems. It is my opinion that since the product of a transpose of a quaternion with a quaternion already has the properties of a norm in a Hilbert space, there is no need to embed quaternions again within another Hilbert space. I like a close shave with Occam's razor.