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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This is a deep and fascinating question — and it’s insightful to sense a connection. While Christoffel symbols and projection operators arise in different mathematical settings (geometry vs. quantum mechanics), they both deal with how structures change or decompose in curved or constrained spaces.

Christoffel Symbols (Geometry)

• Appear in differential geometry.
• Encode how coordinate bases twist and turn in a curved manifold.
• Used to define the covariant derivative.
• Not tensors themselves, but help define how vectors change across space.

Formally:

Projection Operators (Quantum Mechanics)

• Act in Hilbert spaces, often infinite-dimensional vector spaces.
• A projection operator satisfies:

     

• Projects a vector onto a subspace (e.g., eigenstate space of an observable).
• Encodes measurement in quantum mechanics.

Conceptual Bridge

Both structures deal with components of vectors under constraints:

Concept	Christoffel Symbols	Projection Operators
Setting	Curved space/manifold	Hilbert space
Purpose	Adjust derivative to follow curvature	Extract component in a subspace
Key Structure	Covariant derivative	Operator with
Acts On	Geometric vectors/tensors	Quantum states/vectors
Related To	Parallel transport, geodesics	Measurement, observables

Deeper Similarity: Connections and Decompositions

– Christoffel symbols decompose how a vector changes: intrinsic vs. extrinsic parts.
– Projection operators decompose quantum states into subspaces.

Both are types of connections:

• In geometry: a connection tells you how to compare vectors at different points in space.
• In quantum theory: projections (and Berry connections) tell you how quantum states evolve in parameter space.

Advanced View: Gauge Theory Link

In modern physics:

• Christoffel symbols behave like gauge connections for gravity.
• Projection operators appear in Berry phase phenomena and in quantum gauge structures.

In both cases, we talk about how quantities transform when moving around in space or state space.

Connections define how things change when you move around in space (or Hilbert space), and curvature arises when this change depends on the path.

TL;DR

They live in different worlds but share a core idea:

Both Christoffel symbols and projection operators are tools for tracking and constraining change under structure — whether it’s spatial curvature or Hilbert space decomposition.

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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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1. What is Rindler Space?

Rindler space describes the spacetime experienced by an observer undergoing constant acceleration in special relativity. It is useful for understanding:

• Uniformly accelerated motion.
• Event horizons in flat spacetime (analogous to black hole horizons).
• The Unruh effect (where an accelerating observer perceives a thermal bath of particles).

2. Rindler Coordinates: A Natural Frame for Accelerated Observers

In Minkowski spacetime, the metric is:

For an observer moving with constant acceleration \( a \) in the \( x \)-direction, their trajectory satisfies:

Rindler Coordinates \( (\eta, \xi) \)

We define new coordinates \( (\eta, \xi) \) for the accelerated observer:

where:

• \( \eta \) is the proper time of the accelerating observer.
• \( \xi \) is the proper distance from the Rindler horizon.

Rewriting the Minkowski metric in these coordinates gives the Rindler metric:

3. Key Features of Rindler Space

• Only Covers a Portion of Minkowski Spacetime:The Rindler coordinates describe only the right wedge of Minkowski space where \( x > |t| \). The Rindler horizon at \( \xi = 0 \) acts like a black hole horizon.
• Constant Acceleration:In Minkowski space, constant acceleration follows hyperbolic worldlines, which Rindler coordinates naturally describe.
• Analogy with Black Holes:The Rindler horizon behaves like an event horizon, where an accelerating observer experiences the Unruh effect—a thermal radiation due to acceleration.

4. Visual Representations

(a) Rindler Wedge in Minkowski Space

This shows how Rindler coordinates cover only a portion of Minkowski spacetime.

          Minkowski Spacetime
            (t-x diagram)

            |        II (No access) 
            |       
            |------ Rindler Horizon (ξ=0) ------
            |        I (Rindler Wedge)
            |
            ----------------------------------
                      x-axis

(b) Hyperbolic Motion of an Accelerated Observer

This shows how an accelerating observer moves along hyperbolas.

        Worldlines of Accelerating Observers
        ----------------------------------
            \       \       \       \
             \       \       \       \
            --+-------+-------+-------+--> x
             /       /       /       /
            /       /       /       /
        ----------------------------------
                      t-axis

5. Interactive Simulation

For a dynamic visualization of Rindler coordinates and accelerated observers, explore the interactive simulation below:

Rindler Space Interactive Visualization

6. Conclusion

• Rindler space describes the viewpoint of a constantly accelerating observer.
• It is a natural coordinate system for accelerated motion in special relativity.
• It reveals deep connections between acceleration, horizons, and thermodynamics (Unruh effect).

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1. The Event Horizon as a Global Concept

The event horizon is not a physical surface but a mathematical boundary beyond which nothing can escape. Its location depends on all future events in spacetime.

Why is it global?

• The event horizon is defined by whether any signal sent from a point in spacetime can reach a distant observer.
• Since we need to track light paths indefinitely into the future, the event horizon’s position is determined only by considering the entire spacetime evolution.

Light Cones and the Horizon

Light cones show the possible future paths of light. Near a black hole:

• Far from the black hole: light cones are open, allowing escape.
• At the event horizon: the cone tips inward, making escape impossible.
• Inside the event horizon: light cones point entirely toward the singularity.

      Far from the black hole:
          Future light cones open outward
                 |    
                 v    
        \       |       /
         \      |      /
          \     |     /
------------------------------
      Event Horizon (null boundary)
          \     |     /
           \    |    /
            \   |   /
Inside BH:  \  |  /  (light cannot escape)
             \ | /
              \|/
          Singularity

2. The Event Horizon as a Family of Null Geodesics

Null geodesics are paths that light follows through spacetime. The event horizon consists of a family of light rays that just barely fail to escape.

Key Features:

• Outside the horizon, light can escape.
• On the horizon, light is “trapped,” neither falling in immediately nor escaping.
• Inside the horizon, even light is forced toward the singularity.

3. Formation of the Event Horizon

During gravitational collapse, an event horizon forms before the singularity appears. Initially, some light may escape, but once inside the horizon, escape is impossible.

    Stage 1: Star collapsing, some light escapes
       *********    
      *         *    
     *    ⨀    *   --> Some light escapes
      *         *    
       *********

    Stage 2: Event horizon forms
       *********
      *    ●    *   --> Light at horizon is "trapped"
      *         *
       *********

    Stage 3: Black hole fully formed, no escape
       *********
      *         *   --> All paths lead to singularity
      *   ▽     *
       *********

4. Interactive Simulation

For a dynamic visualization of how light behaves near an event horizon, explore the interactive simulation below:

Inside a Black Hole – Interactive Simulation

5. Conclusion

• The event horizon is a global concept: You need to consider all future light paths to define it.
• It consists of null geodesics: Light rays that never escape but also never instantly fall into the singularity.
• It evolves dynamically: The horizon forms before the singularity and expands as mass increases.

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1. How Projection Operators Relate to Group Theory

Projection operators appear in quantum mechanics whenever we have symmetries described by a group \( G \). They help decompose Hilbert spaces into irreducible representations of these groups.

Key Properties:

• Idempotency: \( P^2 = P \) (applying twice is the same as once).
• Orthogonality (for distinct eigenvalues): \( P_i P_j = 0 \) if \( i \neq j \).
• Completeness: The sum of all projectors over a complete basis gives the identity:
  \[
  \sum_i P_i = I.
  \]

2. Projection Operators in Representation Theory

If a quantum system has a symmetry group \( G \), then its Hilbert space can be decomposed into irreducible representations (irreps). The projection operators onto these representations are:

where:

• \( d_\lambda \) is the dimension of the irrep labeled by \( \lambda \).
• \( \chi_\lambda(g) \) is the character of \( g \) in the representation.
• \( U(g) \) is the unitary representation matrix of \( g \).

Visual Representation:

        Group Symmetry in Quantum Mechanics
        --------------------------------------
        | Irrep 1 | Irrep 2 | Irrep 3 | ...
        --------------------------------------
        |   P₁    |   P₂    |   P₃    | ...
        --------------------------------------
            ⬇         ⬇         ⬇
        |ψ⟩ = c₁P₁ + c₂P₂ + c₃P₃

3. Projection Operators in Angular Momentum (SU(2) Symmetry)

In quantum mechanics, the rotation group SO(3) and its double cover SU(2) play a key role.

Decomposing Angular Momentum:

The total angular momentum operator \( J^2 \) commutes with all rotations, meaning its eigenspaces define invariant subspaces. The projection operator onto a definite angular momentum \( j \) is:

Spin-1 Representation of SU(2):

            Angular Momentum Subspaces
        ---------------------------------
        | J=1, m=1 | J=1, m=0 | J=1, m=-1 |
        ---------------------------------
        |    P₊    |    P₀    |    P₋    |
        ---------------------------------
            ⬇          ⬇         ⬇
        |ψ⟩ = aP₊ + bP₀ + cP₋

4. Projection Operators in Parity Symmetry (Z₂ Group)

In systems with parity symmetry, the parity operator \( P \) has eigenvalues \( \pm 1 \). The projection operators are:

Even/Odd Parity States:

        Classical Parity Transformation
        ---------------------------------
        | Even States  (P=+1)  | Odd States  (P=-1) |
        ---------------------------------
        |       P₊        |       P₋        |
        ---------------------------------
            ⬇                    ⬇
        |ψ⟩ = c₊ P₊ + c₋ P₋

5. Projection Operators in Quantum Measurement

In quantum measurement, projection operators describe observable eigenstates and their probabilities follow the Born rule:

Measurement and Decoherence:

        Quantum Measurement and Decoherence
        ---------------------------------------
        | State Before Measurement:  |ψ⟩        |
        ---------------------------------------
        | Projectors:   P₁, P₂, P₃,...            |
        ---------------------------------------
        | Probabilities: P₁⟨ψ|P₁|ψ⟩, P₂⟨ψ|P₂|ψ⟩,... |
        ---------------------------------------

6. Conclusion

• Projection operators decompose Hilbert spaces into irreducible representations.
• They appear in angular momentum (SU(2)), parity symmetry (Z₂), and representation theory.
• They ensure that quantum measurements follow the Born rule and describe state decoherence.

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Read this post on Projection Operators first.

1. What Is Gleason’s Theorem?

Gleason’s theorem states that in a Hilbert space of dimension , the only valid probability measure for quantum measurements must follow the Born rule:

where:

• is the probability of measuring outcome .
• is a **projection operator** representing a measurement.
• is the **density matrix** of the quantum state.

2. Single-Particle Spin Measurement

Consider a spin- particle (like an electron) measured along the -axis.

Spin Observable

The spin operator is:

The possible measured values (eigenvalues) are:

• (Spin up, )
• (Spin down, )

Projection Operators

Each measurement outcome corresponds to a projection operator:

Measurement Probabilities

If the quantum state is , the measurement probabilities are:

Single-Particle Measurement Diagram

       Spin Measurement Device (Stern-Gerlach)
                      |
      ↑ ( +ℏ/2 )      |       ↓ ( -ℏ/2 )
  --------------------->--------------------
        |ψ⟩ = α|+⟩ + β|−⟩

3. Two-Particle Entangled State

Consider two spin- particles in the Bell state:

Observable: Total Spin Along -Axis

The total spin operator is:

The possible measured values are:

• (both particles spin up)
• (both particles spin down)
• (one up, one down)

Measurement Probabilities

For the Bell state , Gleason’s theorem ensures that the measurement outcomes must obey:

Entanglement Measurement Diagram

        Particle A                        Particle B
       -----------                      -----------
       |  +⟩   -⟩ |                      |  +⟩   -⟩ |
       ------------------                 ------------------
                 |  Bell State: |Φ+⟩ = 1/√2 (|+⟩|+⟩ + |−⟩|−⟩)
                 |  
                 |  If A is measured as +ℏ/2, then B must be +ℏ/2.
                 |  If A is measured as -ℏ/2, then B must be -ℏ/2.

4. Conclusion

• Gleason’s theorem proves that quantum measurement probabilities must follow the Born rule.
• Any attempt to assign classical probabilities to measurement outcomes contradicts the additivity condition.
• This rules out non-contextual hidden-variable theories and reinforces the fundamental role of quantum uncertainty.

The post Gleason’s theorem with examples appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Projection Operators and Measurement Outcomes

1. Single-Particle Spin Measurement

Consider a quantum system where a spin-\( 1/2 \) particle (e.g., an electron) is measured along the \( z \)-axis.

Observable: Spin along \( z \)-axis (\( S_z \))

The spin operator \( S_z \) is:

The possible measured values (eigenvalues) are:

• \( +\hbar/2 \) (Spin up, \( |+\rangle \))
• \( -\hbar/2 \) (Spin down, \( |-\rangle \))

Projection Operators:

The corresponding projection operators are:

Measurement Probabilities:

If the quantum state is \( |\psi\rangle = \alpha |+\rangle + \beta |-\rangle \), the probability of measuring \( +\hbar/2 \) (spin up) is:

Similarly, the probability of measuring \( -\hbar/2 \) (spin down) is:

Measurement Process Visualization:

       Spin Measurement Device (Stern-Gerlach)
                      |
      ↑ ( +ℏ/2 )      |       ↓ ( -ℏ/2 )
  --------------------->--------------------
        |ψ⟩ = α|+⟩ + β|−⟩
    

2. Two-Particle Entangled State

Now, consider a system of two entangled spin-\( 1/2 \) particles in the Bell state:

Observable: Total Spin along \( z \)-axis

The total spin operator is:

The possible measured values are:

• \( +\hbar \) (Both particles spin up)
• \( -\hbar \) (Both particles spin down)
• \( 0 \) (One particle spin up, one spin down)

Projection Operators:

For these measurement outcomes, the projection operators are:

Measurement Probabilities:

For the Bell state \( |\Phi^+\rangle \), we calculate:

Entanglement Measurement Visualization:

        Particle A                        Particle B
       -----------                      -----------
       |  +⟩   -⟩ |                      |  +⟩   -⟩ |
       |   |    |  |                      |   |    |  |
       ------------------                 ------------------
                 |  Bell State: |Φ+⟩ = 1/√2 (|+⟩|+⟩ + |−⟩|−⟩)
                 |  
                 |  If A is measured as +ℏ/2, then B must be +ℏ/2.
                 |  If A is measured as -ℏ/2, then B must be -ℏ/2.
    

3. Conclusion

• Single-particle case: Projection operators extract probabilities of spin measurements.
• Two-particle case: Projection operators help analyze entanglement, showing how quantum correlations affect measurement outcomes.

The post Projection Operators along with examples. Gleason’s theorem next appeared first on Time Travel, Quantum Entanglement and Quantum Computing.

Projection Operators and Gleason’s Theorem

1. Projection Operators in Quantum Mechanics

A projection operator \( P \) is a Hermitian operator satisfying:

These operators represent quantum measurement outcomes. If a system is in state \( |\psi\rangle \), the probability of measuring an outcome associated with projection \( P \) is:

2. Projection Operators in Observables

An observable \( A \) with discrete eigenvalues \( a_i \) can be expressed using projection operators \( P_i \):

The probability of measuring \( a_i \) is given by the Born rule:

3. Gleason’s Theorem

Gleason’s theorem states that in a Hilbert space of dimension \( d \geq 3 \), the only possible probability measure satisfying quantum additivity must be:

where \( E \) is a projection operator and \( \rho \) is a density matrix.

4. Implications of Gleason’s Theorem

• Justifies the Born Rule: Probability assignments must follow the standard quantum probability formula.
• Rules out Non-Contextual Hidden Variables: If measurement outcomes are predetermined, the additivity assumption is violated.
• Constrains Deterministic Quantum Theories: No assignment of definite values (0 or 1) to projection operators is consistent with quantum mechanics.

5. Connection to Bell’s Theorem

Gleason’s theorem disproves non-contextual hidden-variable theories but does not rule out contextual hidden-variable theories. Bell later extended this result with Bell’s inequalities, showing that hidden-variable theories must be nonlocal.

6. Conclusion

• Projection operators define quantum measurement outcomes.
• Gleason’s theorem proves that probability in quantum mechanics must follow the Born rule.
• Hidden-variable theories that assume predetermined values contradict quantum probability rules.

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1. Setup of Einstein’s Derivation

Einstein considers two coordinate systems:

• Stationary system : Coordinates
• Moving system : Coordinates , moving at velocity along the -axis.

He assumes that light is emitted from the moving frame’s origin, reflected at some point, and returns to the origin. He also assumes that the transformation equations must be linear.

2. Problem with Infinitesimal Analysis (Misuse of Differentials)

Einstein introduces an infinitesimally small displacement in the moving frame and relates it to time .

Einstein’s Step:

He writes:

since he assumes is independent of time.

Hannon’s Criticism:

• Incorrect Use of Differentials: Einstein treats as independent but also makes it infinitesimally small, a contradiction.
• Incorrect Assumption of Partial Derivative: If were independent of , the correct statement should be: instead of Einstein’s claimed result.

3. Contradiction in Time Transformation

Einstein applies the constancy of the speed of light and sets up the equation:

which follows from the assumption that light takes equal time to travel forward and backward.

Hannon’s Criticism:

• Inconsistent Time Dependence: The derivation implicitly makes a function of , contradicting earlier assumptions.
• Failure to Justify Function Assumptions: The assumed functional form of is not rigorously derived.

4. Flawed Interpretation of Light Signals and Synchronization

Einstein assumes that clock synchronization follows from light signal exchanges, leading to:

Hannon’s Criticism:

• Circular Reasoning: Einstein assumes part of the transformation to derive it.
• Alternative Synchronization Methods: If Einstein’s synchronization method were incorrect, his derivation would be invalid.

5. Final Conclusion from Hannon

Hannon argues that:

• Einstein’s derivation of the Lorentz transformation is invalid.
• The Lorentz transformations may still be correct but require a different derivation.
• The assumptions about differential dependence and clock synchronization need reevaluation.

Counterpoint from Mainstream Physics

Despite Hannon’s criticisms, Einstein’s derivation remains widely accepted because:

• His assumption of linearity is justified by physical symmetry.
• Alternative derivations (such as those based on group theory) confirm the Lorentz transformations.
• Experimental evidence overwhelmingly supports special relativity.

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