In this pioneering paper, P.A.M. Dirac explores the mathematical and physical advantages of using
complex variables in quantum mechanics. Traditional quantum theory relies on wave functions
over real-valued observables (like position q or momentum p). Dirac proposes extending momentum
p into the complex plane, gaining powerful tools from complex analysis—such as
analytic continuation, contour integration, and pole residues.

The key contribution is a “fundamental theorem” about integrals in quantum mechanics: if the integrand
consists of two functions regular (analytic) in complementary half-planes, one can deform the real axis
into a complex contour that avoids singularities without changing the value of the integral.
This idea is applied to wave function representations and operator matrix elements in momentum space.

Dirac applies the theory to familiar problems, notably the hydrogen atom, reformulating
Schrödinger’s solution using this complex variable formalism. This new formulation simplifies
computations and gives deeper insight into boundary conditions, regularity at infinity,
and the origin of quantization.

Detailed Mathematical Explanations

1. Fourier Representation and Complex Momentum

Dirac starts by moving from position q to momentum p space:

\

Unlike standard theory where , Dirac now allows .
If is bounded and vanishes as , then is analytic in
the lower half-plane (Im) and can be analytically continued across the real axis.

2. Dirac’s Fundamental Theorem

In an integral of the form:

\

If:

• is analytic in the upper half-plane
• is analytic in the lower half-plane

then the path of integration can be deformed into the complex plane to avoid poles.

For a simple pole at :

\

The contour is then deformed below the pole, consistent with standard quantum interpretations.

3. Conditions at Infinity

If the Laurent expansion of has a constant term:

\

This implies a delta function in position space:

\

To maintain finite integrals and valid inner products, integration contours must
avoid infinity in the appropriate direction. This leads to a generalization of the
fundamental theorem: if a constant term appears in one factor, the contour must encircle
infinity in the opposite half-plane.

4. Operators in Complex Momentum Space

• Identity operator:
  \

     

• Momentum operator:
  \

     

• Position operator:
  \

     

• Inverse position operator:
  \

     

5. Application to the Hydrogen Atom

Radial Schrödinger equation in position space:

\

is transformed into -representation:

\

Choosing asymptotic behavior:
\

Quantization arises from requiring regularity at poles:
\

Key Takeaways

• Extending to complex values enables analytic tools in QM.
• Integration contours can be deformed to bypass singularities safely.
• Singularities at infinity are handled analogously to real poles.
• Operators acquire elegant matrix forms using complex functions.
• Hydrogen atom solutions are simplified in this formalism.

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A path parametrized by is a way of describing a curve through space by using a single variable, , to trace the position along the path.

The Idea

You have a curve, say, a person walking on a sphere from the equator to the pole. Instead of describing the path just as a set of points , we describe it as a function of a parameter:

This means:

• give the coordinates of a point on the path as changes.
• might represent time, arc length, or an abstract index.

Why Parametrize?

Parametrizing a path lets us:

• Take derivatives along the path: is the tangent vector.
• Track how things like vectors change along the path.
• Write transport equations like .

Analogy: Driving on a Road

– The road is the path.
– is your odometer reading (distance traveled).
– tells you your location at each point.
– gives your direction of motion.

A Math Example

Let’s say you move in a circle:

Then you’re moving along a circle, and is the angle — a natural parameter for this motion.

Summary

When we say “a path parametrized by ,” we mean:

“Here’s a curve through space, and we’ve assigned a smooth way to move along it — so we can differentiate, transport vectors, and do math.”

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Causality in QFT requires that events or measurements that are space-like separated (i.e., events that cannot influence each other) do not affect each other. This is formalized by ensuring that the commutators of local observables vanish for space-like separated points. Specifically, for two local observables O1(x)O_1(x)O1​(x) and O2(y)O_2(y)O2​(y) at spacelike separation, we have:

[O1(x),O2(y)]=0 for (x−y)2<0[O_1(x), O_2(y)] = 0 \text{ for } (x – y)^2 < 0[O1​(x),O2​(y)]=0 for (x−y)2<0

A local observable is an operator defined at a specific point in spacetime or within a local neighborhood. To check for causality, we calculate the commutator of the field operators $\varphi (x)$ and $\varphi (y)$:

$\Delta (x-y)=[\varphi (x),\varphi (y)]$

For free fields, this commutator can be expressed in terms of mode expansions and is shown to vanish for spacelike separations, ensuring causality​​.

Propagators

Propagators in QFT describe the probability amplitude for a particle to travel from one point to another in spacetime. The propagator $D(x-y)$ for a scalar field is defined as:

$D(x-y)=⟨0|\varphi (x)\varphi (y)|0⟩$

This expression gives the amplitude for a particle created at point $y$ to be annihilated at point $x$. Using mode expansions and vacuum expectation values, the propagator can be written as:

D(x−y)=∫d3p(2π)312Epe−ip⋅(x−y)D(x – y) = \int \frac{d^3p}{(2\pi)^3} \frac{1}{2E_p} e^{-ip \cdot (x – y)}D(x−y)=∫(2π)3d3p​2Ep​1​e−ip⋅(x−y)

Interestingly, $D(x-y)$ is non-zero even for spacelike separations, which seems to conflict with causality at first glance. However, this does not violate causality because the commutator of the fields $\varphi (x)$ and $\varphi (y)$ for spacelike separations vanishes due to a cancellation of contributions from particle and antiparticle processes. Specifically:

$[\varphi (x),\varphi (y)]=D(x-y)-D(y-x)$

This ensures that the net effect of any possible propagation between spacelike separated points is zero, preserving causality​​.

In summary, causality in QFT is maintained by ensuring that commutators of field operators vanish for spacelike separations, while propagators provide the amplitude for particle propagation between points in spacetime, contributing to the understanding of how fields and particles interact within the framework of QFT.

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Functionals depend on the ENTIRE space configuration for a system. This implies an inherent non-locality for the state. This should be true for all 3 representations – the Path Integral, the Matrix (Heisenberg) and the Schrodinger representation of states.

1. Path Integral Formulation:
   • In the path integral approach, the transition amplitude between an initial state <katex> ∣ϕi⟩| \phi_i \rangle∣ϕi​⟩ at time tit_iti​ and a final state ∣ϕf⟩| \phi_f \rangle∣ϕf​⟩ at time tft_ftf​ is given by: ⟨ϕf∣e−iH(tf−ti)∣ϕi⟩=∫Dϕ eiS[ϕ]\langle \phi_f | e^{-iH(t_f – t_i)} | \phi_i \rangle = \int \mathcal{D}\phi \, e^{iS[\phi]}⟨ϕf​∣e−iH(tf​−ti​)∣ϕi​⟩=∫DϕeiS[ϕ] where $S[\varphi ]$ is the action functional, and $\mathcal{D}\varphi$ </katex>denotes integration over all field configurations.
2. Schrödinger Representation:
   • In the Schrödinger representation, states are represented as functionals of the field configuration at a given time slice. For a field configuration <katex> $\varphi (\mathbf{x})$, a state $|\Psi ⟩$ is represented by a wavefunctional $\Psi [\varphi ]$: $$\Psi [\varphi ]=⟨\varphi |\Psi ⟩\; </katex>$$
   • Here, <katex>$\varphi (\mathbf{x})$ denotes the field value at each point $\mathbf{x}$ in space.</katex>
3. Non-locality of States:
   • The wavefunctional $\Psi [\varphi ]$ depends on the entire field configuration $\varphi (\mathbf{x})$ over space, not just at a single point. This non-local dependency reflects the fact that the state of the field is determined by its values across the whole spatial domain.
   • For example, in a free scalar field theory, the ground state wavefunctional can be expressed as: Ψ0[ϕ]∝exp⁡(−12∫d3x d3y ϕ(x)K(x−y)ϕ(y))\Psi_0[\phi] \propto \exp \left( – \frac{1}{2} \int d^3x \, d^3y \, \phi(\mathbf{x}) K(\mathbf{x} – \mathbf{y}) \phi(\mathbf{y}) \right)Ψ0​[ϕ]∝exp(−21​∫d3xd3yϕ(x)K(x−y)ϕ(y)) where $K(\mathbf{x}-\mathbf{y})$ is a kernel that encapsulates the spatial correlations of the field.

Non-local Interactions

1. Field Correlations:
   • The functional form of $\Psi [\varphi ]$ indicates that the value of the wavefunctional at a given field configuration depends on correlations between field values at different points in space. This is a direct manifestation of non-locality.
   • Even in the vacuum state, fields at different spatial points are entangled, leading to non-local correlations.
2. Implications for Interactions:
   • When interactions are included (e.g., in an interacting scalar field theory with a ϕ4\phi^4ϕ4 term), the wavefunctional becomes even more complex, with non-local dependencies reflecting the interaction terms in the Hamiltonian.
   • The path integral formulation sums over all possible field configurations, accounting for all possible interactions, which are inherently non-local because the action $S[\varphi ]$ typically includes spatial integrals of interaction terms.
3. Example: Instantaneous Propagation:
   • In the Schrödinger picture, the evolution of states can be influenced by the entire field configuration instantaneously, implying non-local effects. For example, the potential energy in a scalar field theory involves integrals over the entire space, indicating that the interaction energy at a point depends on the field values everywhere.

Summary

In QFT, states represented as functionals of field configurations exhibit non-locality because these functionals depend on the entire spatial field configuration. This non-locality is inherent in the path integral formulation, where the action includes integrals over space, and interactions are accounted for by summing over all possible field configurations. Thus, the non-local nature of states in QFT reflects the fundamental structure of the theory, where interactions and correlations extend across the entire spatial domain.

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• Frequency is related to the energy of the photon. Higher frequency photons have higher energy and vice versa.
• Low energy photons (such as radio photons) behave more like waves, while higher energy photons (such as X-rays) behave more like particles.

What happens in High Energy Collisions? Are photons powerful enough to form electron positron pairs?

In high energy collisions, photons are emitted. Sometimes, the energy of the photons is large enough that it can b e

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EXTERNAL to the object – is key to understanding relativity.

In a speeding spaceship, little of interest happens INSIDE the spaceship. Time flows at the usual rate, and lengths measure their usual selves.

It is only when viewed from OUTSIDE the spaceship, that things start looking weird.

Taken to an extreme, if an object is indeed moving at light speed, it will appear completely stationary (frozen in time) to an external observer (as predicted on a black hole’s event horizon)

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Energy to knock an atom out

Consider a Helium Atom with it’s two outer electrons. To knock an electron, one needs a very specific, exact amount of energy (corresponding to a laser wavelength of 50 nm). If you deviate even 0.0000001 % from this wavelength, the electron will not be knocked out. Classically – say you have to knock pluto out of it’s orbit. Whether you hit it with a small missile, a medium missile or a large missile – each missile will have some effect on the orbit. The larger the missile, the greater the effect. Not so in QM. Only the EXACT size missile will work.

The Probability Wave (Wave Function) – Is it real or not?

In some cases, it is an actual real wave!

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