Lesson 2
Phase as Continuous Rotation
How changing phase produces smooth rotational motion.
In QM1, relational phase transport is represented mathematically by continuous rotations. As the phase changes smoothly, every point rotates smoothly around the origin. What ordinary quantum mechanics calls evolution is being reconstructed here as relational phase transport.
Why Continuity Matters
Once relational transport is available, QM1 asks what mathematical structure is required when phase transport varies continuously.
Continuity prevents arbitrary jumps between nearby transport states and allows the phase structure to be represented by continuous transformations.
Representation Assumptions
Before any rotation can be drawn, QM1 states four explicit assumptions. They are not derived from the PDT0 axioms — they are the conditions under which the reconstruction proceeds.
R1a
Phase configurations embed into a real vector space.
R1b
That space is finite dimensional.
R1c
Phase acts continuously and linearly.
R2
Different phase angles produce different transformations (faithfulness).
These assumptions are introduced explicitly. QM1 investigates what follows if they are true.
Auto Rotate
Keep rotating continuously
Live Values
- Current Phase
- θ = 0.0°
- Current Position
- (1.00 , 0.00)
Real World Connection
- Clock hands
- Spinning wheels
- Planetary motion
- Rotating electromagnetic fields
QM1 borrows the mathematics of rotation — not these physical objects themselves. The turning is abstract: it describes relationships, not spinning matter.
Phase Composition
Two rotations make one
Watch A, then B, then the single equivalent rotation that lands in exactly the same place.
What Have We Learned?
Continuous phase behaves like continuous rotation. Instead of jumping between states, every small change in phase produces a small change in orientation. This smooth behaviour is one of the key mathematical foundations developed in QM1.
The next lesson asks how small a space can carry this rotation: a faithful continuous real representation of continuous rotation requires two dimensions.
Learning Check
If phase changes smoothly, how does the vector move?
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