Dirac’s Transformation Theory

Matrix mechanics and wave mechanics first looked like different theories. Matrix mechanics used arrays of transition quantities; wave mechanics used differential equations for wavefunctions. Dirac’s transformation theory helped reveal a deeper unity: both are representations of the same quantum structure.

In modern language, a pure state is not identical with any one column vector or wavefunction. A representation is a way of writing the state relative to a chosen basis or complete set of observables. Dirac’s transformation theory made that idea central.

Matrix and Wave Representations

By 1926, quantum mechanics had several successful languages:

• Heisenberg’s matrix mechanics emphasized transition quantities and noncommutative products.
• Born and Jordan’s formulation clarified the matrix algebra and canonical commutators.
• Schrödinger’s wave mechanics described systems by wavefunctions and differential equations.
• Born’s probability interpretation connected amplitudes to observed frequencies.

The urgent problem was not only to show that these languages gave the same spectra in examples. It was to understand why they were the same theory.

Dirac’s contribution was to treat transformations between descriptions as part of the theory itself. Instead of asking whether the matrix or wave picture was more real, transformation theory asks how to pass between representations without changing the underlying physical state.

Pascual Jordan developed a closely related transformation theory independently. His 1927 paper explicitly connected matrix mechanics, wave mechanics, and Dirac’s q-number theory through statistical postulates. The modern account below uses later notation to explain their shared idea.

Transformation Theory

In finite-dimensional modern notation, let $\{\lvert a\rangle\}$ and $\{\lvert b\rangle\}$ be two orthonormal bases. A state $\lvert\psi\rangle$ has components

$$\psi_a = \langle a\vert\psi\rangle, \qquad \psi_b = \langle b\vert\psi\rangle.$$

The overlap between bases is the transformation amplitude

$$S_{ba} = \langle b\vert a\rangle.$$

Then

$$\psi_b = \sum_a \langle b\vert a\rangle \psi_a.$$

The same idea works for continuous representations, with sums replaced by integrals. Position and momentum wavefunctions are related by a transformation kernel:

$$\psi(x) = \langle x\vert\psi\rangle, \qquad \phi(p) = \langle p\vert\psi\rangle.$$

With a common convention,

$$\langle x\vert p\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ipx/\hbar},$$

so

$$\psi(x) = \int_{-\infty}^{\infty} \langle x\vert p\rangle \phi(p)\,dp.$$

The kernel depends on the two bases, not on the prepared state. Here the generalized momentum states satisfy $\langle p\vert p^\prime\rangle=\delta(p-p^\prime)$; they are not normalizable vectors. The Fourier transform is unitary on square-integrable wavefunctions.

Dirac’s transformation viewpoint separates the abstract state from its representations. The two component lists describe the same state. The overlap matrix relates the bases; the observable matrix must change with the same convention.

States, Representations, and Observables

Transformation theory points toward the modern state-observable language. A state can be represented in the eigenbasis of an observable $A$, in the eigenbasis of another observable $B$, or in a continuous position or momentum representation. The physical state is the invariant object behind those coordinate descriptions.

For a discrete nondegenerate observable $A$, the modern spectral form is

$$A = \sum_a a\, \lvert a\rangle\langle a\rvert.$$

The amplitude for finding outcome $a$ in state $\lvert\psi\rangle$ is

$$\langle a\vert\psi\rangle,$$

and the probability is

$$P(a) = \lvert\langle a\vert\psi\rangle\rvert^2.$$

For another observable $B$, the relevant amplitudes are instead $\langle b\vert\psi\rangle$. The transformation amplitudes $\langle b\vert a\rangle$ tell how the two descriptions are related.

The observable must be transformed along with the state. With the convention $S_{ba}=\langle b\vert a\rangle$ used above,

$$\psi^{(b)}=S\psi^{(a)},\qquad A^{(b)}=S A^{(a)}S^\dagger.$$

Since $S^\dagger S=I$, the expectation value remains unchanged. A change of coordinates does not by itself prepare a new state or perform a measurement.

Bridge to Bra-Ket Notation

Dirac’s later bra-ket notation packages the transformation viewpoint compactly:

• a ket $\lvert\psi\rangle$ denotes the state vector;
• a bra $\langle a\rvert$ extracts an amplitude in the $a$ representation;
• an inner product $\langle a\vert\psi\rangle$ is a component or transition amplitude;
• an outer product $\lvert a\rangle\langle a\rvert$ is a projector in the nondegenerate discrete case.

Historically, the notation and the conceptual framework matured over time. It is still useful to read the notation as a compact summary of transformation theory: kets are not column vectors until a basis is chosen, and bras are not merely decorative row symbols. They encode the dual pairing that produces amplitudes.

For the working convention, see Bra-Ket Notation. For the mathematical translation, see Dirac Notation as Linear Algebra.

Continuous and Infinite-Dimensional Representations

Continuous bases require generalized eigenvectors and distributions; unbounded operators require domains. Within a fixed Hilbert space, the complete orthonormal bases used above are related unitarily. For infinite systems and quantum fields, however, different representations of an observable algebra need not be unitarily equivalent. That broader question is not settled by the finite-matrix calculation.

Common Mistakes

• Treating the wavefunction as the state itself rather than a representation of the state.
• Treating a matrix as the operator itself without specifying the basis.
• Thinking transformation theory is only about time evolution. It is also about changing representation.
• Forgetting that $\langle b\vert a\rangle$ is an amplitude, not a classical conditional probability.
• Applying finite-dimensional basis formulas to continuous spectra without distributional care.
• Reading modern bra-ket notation back into every 1920s paper as if it was already standardized.

Exercises

1. Let $\{\lvert a\rangle\}$ and $\{\lvert b\rangle\}$ be orthonormal bases with $S_{ba}=\langle b\vert a\rangle$. Show that $\psi_b=\sum_a S_{ba}\psi_a$.

Solution

Insert the identity in the $a$ basis:

$$I = \sum_a \lvert a\rangle\langle a\rvert.$$

Then

$$\psi_b = \langle b\vert\psi\rangle = \sum_a \langle b\vert a\rangle \langle a\vert\psi\rangle = \sum_a S_{ba}\psi_a.$$

2. A normalized state is $\lvert\psi\rangle=(\lvert0\rangle+i\lvert1\rangle)/\sqrt2$. Find its components in the basis $\lvert\pm\rangle=(\lvert0\rangle\pm\lvert1\rangle)/\sqrt2$ and check normalization.

Solution

$$\langle+\vert\psi\rangle=\frac{1+i}{2},\qquad \langle-\vert\psi\rangle=\frac{1-i}{2}.$$

Both probabilities are $1/2$, so their sum is one. The components have changed, but the state has not. Squaring the old components before transforming would discard the relative phase needed to calculate probabilities in another basis.

3. Why is $\langle b\vert a\rangle$ not a classical conditional probability?

Solution

It is generally complex and can interfere with other amplitudes. A probability is obtained only after the relevant amplitudes have been combined and squared according to the Born rule. The quantity $\langle b\vert a\rangle$ is therefore a transformation amplitude, not an ordinary conditional probability.

4. Explain why calling $\psi(x)$ “the state” can be misleading.

Solution

$\psi(x)=\langle x\vert\psi\rangle$ is the position representation of the abstract state $\lvert\psi\rangle$. The same state can also be represented by momentum-space amplitudes, energy-basis components, or another basis. Calling $\psi(x)$ the state hides the basis dependence and can make representation changes look like physical changes.

References

• P. A. M. Dirac, “The fundamental equations of quantum mechanics,” Proceedings of the Royal Society A 109, 642-653, 1925, DOI: 10.1098/rspa.1925.0150.
• P. A. M. Dirac, “The physical interpretation of the quantum dynamics,” Proceedings of the Royal Society A 113, 621-641, 1927, DOI: 10.1098/rspa.1927.0012.
• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• M. Jammer, The Conceptual Development of Quantum Mechanics, 2nd ed., American Institute of Physics, 1989.
• P. Jordan, “Über eine neue Begründung der Quantenmechanik,” Zeitschrift für Physik 40, 809–838, 1927, DOI: 10.1007/BF01390903.
• J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Springer, 1982-2001.
• B. L. van der Waerden, ed., Sources of Quantum Mechanics, Dover, 1968.