The Action and Path Integral in Quantum Mechanics

Core Concept Overview

This framework connects classical and quantum mechanics through the concept of the action and extends it via the path integral, which provides a complete reformulation of quantum theory.

1. The Classical Action (S)

In classical mechanics, the action is a fundamental quantity within the Lagrangian formulation. It is a functional—a function of a function—that assigns a single real number to any conceivable path q(t) between two points in configuration space.

S[q(t)] = ∫_t1^t₂ L(q, ˙q, t) dt

Here, L is the Lagrangian (typically Kinetic Energy minus Potential Energy: T - V), q(t) is the generalized coordinate (like position), and ˙q(t) is its time derivative (velocity).

Principle of Least Action (Hamilton's Principle)

The actual path taken by a classical particle between two points is the one that makes the action stationary (typically a minimum). This variational principle yields the Euler-Lagrange equations of motion, which are mathematically equivalent to Newton's laws.

Classical Summary: Nature selects the one unique path that extremizes the action.

2. The Quantum Path Integral (Feynman's Formulation)

Richard Feynman revolutionized quantum mechanics by reinterpreting the action. In the quantum domain, determinism dissolves.

Core Quantum Idea

A quantum particle does not take a single, definite path between an initial point A and a final point B. Instead, it theoretically explores every possible path simultaneously. Each path contributes to the total quantum amplitude for the transition from A to B.

The Path Integral Mechanism

Step 1: Amplitude per Path
To each hypothetical path q(t), we assign a complex phase factor:

Phase Factor = e^iS[q(t)]/ħ

Here, S[q(t)] is the classical action for that specific path, and ħ is the reduced Planck constant. The magnitude of this factor is always 1; only its phase (angle in the complex plane) changes, dictated by the action.

Step 2: Sum Over All Paths (The Integral)
The total probability amplitude K(A → B) (the propagator) is found by summing (integrating) this phase factor over all paths connecting A and B:

K(A → B) = ∫_{all paths q(t)} 𝒟q(t) e^iS[q(t)]/ħ

The symbol ∫ 𝒟q(t) represents a functional integral—an infinite-dimensional integral over all possible functions (paths).

Step 3: Quantum Interference
Paths with very different actions have wildly different phases and tend to interfere destructively (cancel out). Paths where the action is stationary (i.e., the classical path and its neighbors) have nearly identical phases and interfere constructively.

Step 4: From Amplitude to Probability
The probability for the particle to go from A to B is the absolute square of the total amplitude:

P(A → B) = |K(A → B)|²

Key Implications

Emergence of Classical Physics: The classical path of least action is the path of stationary phase. In the limit where ħ → 0 (the classical limit), constructive interference is infinitely sharp, isolating only the classical path.

Double-Slit Explained: The path integral naturally accounts for a particle going through both slits. The amplitude sums over the path through the left slit and the path through the right slit; their interference creates the observed pattern.

Advantages: This formulation is conceptually elegant, makes symmetries transparent, and generalizes seamlessly to quantum field theory, where one sums over all possible field configurations.

Analogy: The Drunkard's Walk

Classical (Sober Walk): A person takes the single, shortest, most efficient route from the pub to home.

Quantum (Drunkard's Walk): Imagine a profoundly drunk person who, in a sense, stumbles along every conceivable zigzag path at once. For each path, we attach a spinning arrow (the phase e^iS/ħ). Adding all arrows, most cancel (pointing in random directions). Arrows for paths similar to the sober walk point in nearly the same direction and reinforce each other. Thus, the highest probability concentrates near the classical path.

Summary: Classical vs. Quantum View

Concept	Role in Classical Mechanics	Role in Quantum Mechanics (Path Integral)
Action (S)	A number to be minimized. It selects the one true path.	A number determining the quantum phase eiS/ħ for each possible path.
Path	A single trajectory q(t) obeying deterministic laws.	All possible trajectories q(t) connecting the two points.
Core Principle	Principle of Least Action.	Sum over all histories/paths, weighted by eiS/ħ.
Outcome	Deterministic trajectory.	Probability amplitude, from which observable probability is derived.

In Essence

The action is the fundamental quantity that dictates the quantum phase. The path integral is the rule for summing these phases over all conceivable paths to calculate quantum probabilities. It reveals quantum mechanics as a theory of "everything that might have happened" (Feynman).

Quantum Error Correction Threshold Achievement

According to the latest research progress, a team from the University of Science and Technology of China (USTC) achieved quantum error correction below the threshold in the surface code regime for the first time at the end of 2025. The core of this breakthrough is not a single specific numerical value, but a critical technical state: the overall error rate of the system's physical qubits has dropped below the theoretical limit required for the surface code scheme to provide positive error correction, realizing the state of "error suppression."

Research Details and Technical Comparison

For clarity, here is a comparison of the key information between the Chinese team (USTC) and Google in achieving this milestone:

Research Aspect	University of Science and Technology of China (USTC)	Google
Research Team	Jianwei Pan, Xiaobo Zhu, Chengzhi Peng, Fusheng Chen, et al.	Google Quantum AI Lab
Publication Date	December 2025	Early 2025 (Nature paper)
Experimental Platform	Zuchongzhi 3.2 (107 qubits)	Willow (105 qubits)
Error-Correcting Code	Distance-7 surface code	Distance-7 surface code
Key Metric	Error Suppression Factor: 1.40	Error Suppression Factor: 2.14
Technical Approach	All-microwave quantum state leakage suppression architecture	DC-pulse quantum state leakage suppression method
Core Advantage	Fewer hardware constraints, lower wiring complexity, greater potential for scalability	Higher error suppression factor

Note on Error Suppression Factor (Λ): An Error Suppression Factor (Λ) greater than 1 means the logical error rate decreases exponentially as the code size increases. This is direct experimental evidence that the system is operating below the error correction threshold.

Understanding the "Error Correction Threshold"

Simply put, the error correction threshold is like a "passing line":

• Physical qubit error rate above the threshold: The additional errors introduced by the correction process itself outweigh the benefits, leading to "more errors with correction."

• Physical qubit error rate below the threshold: Error correction yields a net positive benefit. The system enters the ideal "error suppression" state, where a logical qubit can be more stable than any of its constituent physical qubits.

Key Differences in Technical Approaches

While both teams achieved this milestone, their technical paths differ significantly:

China's "All-Microwave" Path

Uses microwave signals for unified control, reuses existing hardware, and is naturally suited for multiplexing. This greatly reduces wiring complexity and hardware overhead in the extreme low-temperature environment required for large-scale expansion.

Google's "DC-Pulse" Path

Suppresses errors by applying DC pulses, which is effective but imposes specific constraints on chip design (e.g., qubit connectivity) and incurs greater hardware resource overhead during large-scale scaling.

In summary, while Google's solution currently demonstrates a better specific metric (error suppression factor), the Chinese team's approach is architecturally simpler and is considered to have greater potential for scalability on the path toward million-qubit-scale quantum computers.

In short, the USTC team is the second in the world, after Google, to achieve sub-threshold quantum error correction in the surface code regime. This breakthrough is a critical watershed that quantum computing must cross to move from laboratory prototypes to practical applications, and the proposed new architecture provides an important technical option for future large-scale scaling.

If you are interested in the basic principles of quantum error correction or why surface codes are the current mainstream approach, further explanations are available.

BYD vs. Tesla: 2025 Electric Vehicle Production & Sales

Yes, BYD has surpassed Tesla in pure electric vehicle (BEV) production and sales for the 2025 calendar year. This marks the first full year BYD has taken the top spot from Tesla, which had been the leader for years.

2025 Annual Performance & Market Position

🏆 BYD (2025)

Pure Electric (BEV) Sales 2.26 million

Total Vehicle Sales (BEV + PHEV) 4.55 million

Key Market Global leader; Strong growth in Europe

Current Position World's largest EV manufacturer

🔋 Tesla (2025)

Pure Electric (BEV) Sales 1.64 million

Total Vehicle Sales (BEV + PHEV) 1.64 million (BEV only)

Key Market Global; Facing challenges in core markets

Current Position Previously the world's largest

🔍 Key Factors Behind the Shift

The change in leadership results from a combination of different trajectories for the two companies in 2025:

BYD's Global Growth

BYD achieved a 27.9% year-on-year increase in BEV sales.

A major driver was its overseas success, with exports surging by 150.7% to over 1 million vehicles.

Despite a challenging market in China, BYD's global expansion, including new factories in places like Hungary, helped secure the top spot.

Tesla's Sales Decline

Tesla's annual sales fell by approximately 9% in 2025.

Political and Brand Factors: Elon Musk's political activities and alignment with the Trump administration are cited as having alienated some customers and negatively impacted the brand. A Yale University study suggested this could have significantly reduced Tesla's sales potential.

Policy Changes: The withdrawal of U.S. federal EV subsidies under the Trump administration hurt demand.

Product Transition: Tesla's production was affected by the ramp-down of the old Model Y and the ramp-up of its successor, making the popular model unavailable for months.

📊 A Look at the Broader Picture

Defining the "Largest": When comparing, the term "largest EV maker" typically refers only to Battery Electric Vehicles (BEVs). If you include Plug-in Hybrid Electric Vehicles (PHEVs) in the count, BYD has been the overall "New Energy Vehicle" leader for several years due to its strong hybrid lineup.

Profit vs. Volume: Despite selling fewer cars, Tesla has historically been the far more profitable company. A comment on one industry report notes that Tesla still makes the vast majority of global EV profits, which funds its other ventures like energy storage and robotics.

Looking Ahead: For 2026, BYD aims to sell 1.6 million vehicles outside China, while Tesla's sales are forecast to recover slightly to around 1.75 million. However, analysts widely expect the intense competition between these two, and with other global automakers, to continue.

Data reflects 2025 calendar year production and sales figures. The comparison focuses on Battery Electric Vehicle (BEV) volumes for the "world's largest EV maker" title.

Understanding Hypotenuse, Adjacent, and Opposite Sides

Important: Adjacent and opposite are sides of a right triangle, defined relative to a specific acute angle. The hypotenuse is fixed.

1. The Hypotenuse

Definition

The longest side of a right triangle.

Location

It is always the side opposite the right angle (90° angle).

Key Fact

It never changes for a given triangle and is always the hypotenuse, no matter which acute angle you're using as your reference.

2. Adjacent Side (Relative to a chosen angle)

Definition

The leg that forms the chosen acute angle, along with the hypotenuse.

Memory Aid

The side touching or next to the angle (other than the hypotenuse).

3. Opposite Side (Relative to a chosen angle)

Definition

The leg that is across from the chosen acute angle. It does not form the angle.

Memory Aid

The side facing the angle.

Visual Explanation

View from Angle θ (Theta)

θ

Hypotenuse

Adjacent (to θ)

Opposite (to θ)

• Hypotenuse: The slanted side (always)
• Adjacent: The bottom horizontal leg (touching θ)
• Opposite: The vertical leg (across from θ)

View from Angle α (Alpha)

α

Hypotenuse

Adjacent (to α)

Opposite (to α)

• Hypotenuse: The same slanted side (unchanged)
• Adjacent: The vertical leg (now touching α)
• Opposite: The bottom horizontal leg (now across from α)

Notice: The Opposite side for θ is the Adjacent side for α, and vice-versa. The sides swap roles when you change reference angles!

Connection to Trigonometry

This naming convention is the foundation of the three primary trigonometric ratios:

Function	Ratio	Explanation
Sine (sin)	Opposite / Hypotenuse	Compares the side opposite the angle to the hypotenuse
Cosine (cos)	Adjacent / Hypotenuse	Compares the side adjacent to the angle to the hypotenuse
Tangent (tan)	Opposite / Adjacent	Compares the side opposite to the side adjacent to the angle

Example: In the first triangle above, for angle θ:

• sin θ = (Opposite to θ) / Hypotenuse
• cos θ = (Adjacent to θ) / Hypotenuse
• tan θ = (Opposite to θ) / (Adjacent to θ)

SOH-CAH-TOA
(The classic mnemonic for remembering trigonometric ratios)

SOH
Sine = Opposite / Hypotenuse

CAH
Cosine = Adjacent / Hypotenuse

TOA
Tangent = Opposite / Adjacent

Key Takeaway

Always ask: "Which acute angle am I using as my reference point?" Once you pick the angle:

• Hypotenuse is fixed (opposite the right angle).
• Opposite is the side directly across from your chosen angle.
• Adjacent is the side next to your angle that isn't the hypotenuse.

This HTML page visually explains the concepts of hypotenuse, adjacent, and opposite sides in right triangles.

What is More Important: Radians or Degrees?

Radians are fundamentally more important for mathematics and physics, while degrees are more intuitive for everyday life.

Think of it this way: Radians are the "native language" of angles, built into the very structure of math. Degrees are a convenient, human-made translation.

The Case for Radians (Why They Are More Important)

Natural Connection to Circles

One radian is defined as the angle created when you take the radius of a circle and wrap it along the circumference. The formula for arc length becomes beautifully simple: Arc Length = Radius × Angle (in radians), or s = rθ. This formula doesn't work cleanly with degrees without a conversion factor.

Calculus & Higher Math Works Beautifully

This is the most critical reason. The derivative of sin(x) is cos(x) only if x is in radians. Taylor series expansions and other advanced mathematical tools only work naturally when angles are measured in radians. They are the "natural unit" that makes the math of waves, oscillations, and growth clean and elegant.

They Are Unitless

A radian is a ratio of two lengths (arc length / radius), so it has no dimension. This makes it seamlessly integrable into physics formulas, like angular velocity (ω = θ/t).

Universal in Science and Engineering

Advanced fields like physics, engineering, and computer graphics exclusively use radians. To understand signal processing, orbital mechanics, or quantum physics, you must use radians.

The Case for Degrees (Why They Persist)

Human Intuition

The base-360 system is highly divisible (by 2, 3, 4, 5, 6, 8, 9, 10, 12...), which is excellent for mental estimation and simple geometry. A right angle (90°) is easy to visualize and communicate.

Historical & Cultural Pervasiveness

Degrees have been used for millennia in navigation, construction, and basic geography. They are the first unit of angle measurement most people learn.

Practical for Simple Tasks

For telling time (360° for a clock face), reading a compass (bearing 45°), or cutting a pie, degrees are perfectly adequate and intuitive.

Analogy: Temperature

Degrees are like Fahrenheit or Celsius – practical for everyday use ("it's 70°F outside").
Radians are like Kelvin – the absolute, scientific scale where fundamental physical laws work simply and directly.

The Verdict

For calculation, theory, and advanced STEM fields: Radians are unquestionably more important. They are the correct and natural unit.

For communication, basic geometry, and everyday life: Degrees are more common and intuitive.

How to think about it: You need to be bilingual. Learn to think in both, but understand that radians are the language in which the universe's mathematical laws are most simply written. When in doubt in a technical or mathematical context, use radians.

Systems Framework: From Natural Epistemology to Artificial Intelligence

A structured architecture mapping human cognition to AI systems through properties, parameters, and attributes.

Core Philosophical Framework: From Natural to Artificial

Natural Epistemology (Human): Senses + Intelligence + Objects → Perception & Knowledge

Artificial Intelligence (System): Sensors + Algorithms + Data → Models & Actions

Key Insight: Properties, Parameters, and Attributes serve as the formal, quantifiable representations of the qualities perceived by senses and reasoned about by intelligence.

Systems Architecture: A Three-Layer Model

This model maps directly to the hardware, firmware, software paradigm, creating a coherent information pipeline.

Layer 1: Hardware Layer (The "Body" & Raw Interface)

Correlate to:

The Five Senses + Physical Objects.

Role:

Transduces physical phenomena (objects, events) into structured digital data.

Incorporating Properties, Parameters, Attributes:

Attributes (Intrinsic to Objects/Signals): These are the raw, measurable qualities of the physical world.

Example (Vision): Pixel luminance (brightness), wavelength (color), spatial coordinates.

Example (Audio): Frequency, amplitude, phase.

Example (Touch Sensor): Resistance, capacitance, pressure (psi).

System Format: These are low-dimensional, physically-grounded data vectors from sensors. They are the atomic primitives of the system's perception. A parameter here might be the sampling rate (firmware-defined) that governs how these attributes are captured.

Layer 2: Firmware / Middleware Layer (The "Perceptual Spine")

Correlate to:

Lower-level, quasi-reflexive perception and signal processing (the "hardwired" parts of intelligence).

Role:

Converts low-level attributes into higher-level properties and features. This layer performs invariant detection and filtering.

Incorporating Properties, Parameters, Attributes:

Properties (Derived & Relational): These are computed interpretations of combined attributes.

Example: From pixel attributes (color, brightness), compute the property texture = {"rough", "smooth"} or edge_strength = 0.87.

Example: From audio attributes (frequencies), compute the property pitch = 440Hz or phoneme = "/ae/".

Parameters (The Tunable Knobs): This layer is parameter-heavy. These are the fixed or tunable settings that control how attributes are synthesized into properties.

Examples: Edge detection kernel coefficients, filter cut-off frequencies, time-window sizes, noise-floor thresholds, activation functions in a neural net layer.

System Format: A pipeline of parameterized transforms (e.g., DSP filters, convolutional kernels, spectral analyzers). Its output is a feature vector—a structured set of properties ready for cognitive software.

Layer 3: Software / Cognitive Layer (The "Mind")

Correlate to:

Higher-order Intelligence + Synthesis.

Role:

Uses properties and features from the firmware layer to form abstract representations, make decisions, learn, and act. This is where epistemology becomes explicit.

Incorporating Properties, Parameters, Attributes:

Attributes (in the software sense): Now become symbolic or semantic labels attached to conceptual objects.

Example: An object in a knowledge graph has attributes: {"type": "cat", "size": "medium", "affectionate": True}. These are high-level assertions.

Properties are used as evidence to assign these attributes via classification (if "furry" and "meows" then type:cat).

Parameters (The Learned & Adaptive Core): These are the learnable weights of models (e.g., weights in a Deep Neural Network, probabilities in a Bayesian network, rule weights in an expert system).

These parameters encode the system's epistemology—its "beliefs" about how sensory properties correlate with conceptual attributes and categories. They are updated via learning algorithms.

System Format: Models (e.g., neural networks, probabilistic graphs, symbolic KBs) defined by:
1. Architecture/Logic (the fixed structure of reasoning).
2. Parameters (the malleable knowledge within that structure).
3. Input/Output Schemas (mapping perceptual properties to cognitive attributes and actions).

Unified Systems View: The Information Flow

Natural Epistemology Component	AI System Component	Form of Representation (Properties, Parameters, Attributes)
Object in the World	Data Source / Target	Has physical attributes (mass, reflectivity, etc.).
Sense Organ (e.g., Eye)	Hardware Sensor (e.g., Camera)	Outputs signal attributes (pixel arrays). Governed by physical parameters (exposure, gain).
Perceptual Processing	Firmware/Middleware Layer	Transforms signal attributes into perceptual properties (edges, textures). Uses algorithmic parameters (filter coefficients).
Intelligence (Understanding)	Software/Cognitive Layer	Maps properties to semantic attributes (identity, intent, risk). Uses model parameters (neural weights) refined by learning.
Knowledge	Internal Model State	A structured network where entities have attributes and relations have properties. The model's parameters are the encoded knowledge.
Action / Expression	Actuators & Outputs	Commands defined by control parameters, which are functions of the system's state (attributes + properties).

Pedagogical Value of This Framework

Demystifies AI

It shows AI not as magic, but as a systematic engineering implementation of the natural process of knowing.

Clarifies Terminology

Attribute: A qualifier. Can be low-level (sensor data) or high-level (semantic label).

Property: A descriptive characteristic derived from relationships or computations. It often sits between raw data and abstract knowledge.

Parameter: A system variable that controls a process. It can be fixed (design choice), tunable (knob), or learned (the essence of AI).

Emphasizes the Pipeline

Students see that intelligence is built on a layered transformation of representations, each with its own type of parameters and attributes.

Unifies Symbolic and Sub-Symbolic AI

High-level symbolic attributes (e.g., dangerous) can be grounded in sub-symbolic properties (e.g., rapid_looming_motion = true) via parameterized models.

In essence: By adopting this systems format, you teach that building an AI is the process of designing a pipeline that transforms physical attributes into cognitive attributes, mediated by parameters that are either engineered or learned. This perfectly captures the transition from natural epistemology to artificial intelligence.

Systems Framework for AI Education | Natural Epistemology to Artificial Intelligence

NASA's Dragonfly Mission to Titan

🚀 ACTIVE DEVELOPMENT | LAUNCH: 2028 | ARRIVAL: 2034

🚀 Mission Overview & Status

Primary Objective Investigate Titan's prebiotic chemistry and habitability to understand the origins of life.

Launch Window Planned for July 5-25, 2028.

Titan Arrival Scheduled for 2034.

Mission Duration 3.3 years of surface operations.

Current Status In development and testing; passed its Critical Design Review in April 2025, authorizing full-scale construction.

🛰️ The Spacecraft: An Innovative Titan Flyer

Dragonfly is an octocopter—a rotorcraft with eight rotors, roughly the size of a small car. It leverages Titan's unique environment:

Key Environmental Advantages

Dense Atmosphere Four times denser than Earth's, making flight efficient.

Low Gravity About 1/7th of Earth's, reducing the power needed to fly.

Power Source A Multi-Mission Radioisotope Thermoelectric Generator (MMRTG). This nuclear battery recharges Dragonfly's lithium-ion batteries during the 8-Earth-day-long Titan night.

🪐 Why Titan? A World of Prebiotic Chemistry

Titan is a high-priority target for astrobiology because it resembles a frozen version of early Earth.

Organic Rich Its atmosphere and surface are filled with complex, carbon-rich molecules—the building blocks of life.

Active "Hydrological" Cycle Features clouds, rain, rivers, and lakes of liquid methane and ethane, similar to Earth's water cycle.

Subsurface Ocean Evidence suggests a global saltwater ocean beneath its icy crust, a potential habitat.

🔬 Scientific Goals & Instruments

Dragonfly will fly to dozens of distinct locations across Titan to sample and analyze surface materials. Its instruments will:

Analyze Surface Composition Use drills and a mass spectrometer (DraMS) to identify organic molecules.

Probe Beneath the Surface A gamma-ray and neutron spectrometer (DraGNS) will detect subsurface elements.

Monitor Environment A geophysics and meteorology package (DraGMet), including a seismometer provided by JAXA, will measure weather and "Titanquakes".

Scout & Image A suite of cameras (DragonCam) will capture aerial and microscopic images.

🗺️ The Flight Plan: An Epic Journey

Dragonfly will embark on an ambitious aerial expedition:

Initial Landing Touchdown in the Shangri-La dune fields, similar to linear dunes on Earth.

"Leapfrog" Exploration It will perform short flights, eventually building up to journeys of up to 5 miles (8 km) per hop.

Final Destination The mission aims to reach the Selk impact crater, where past liquid water likely mixed with organic material.

Total Travel Over its mission, Dragonfly is expected to fly more than 108 miles (175 km), vastly exceeding the range of Mars rovers.

⚙️ Recent Progress & Challenges

The mission is making tangible progress but has faced hurdles:

Recent Milestones Successful testing of rotors in Titan-like conditions, delivery of flight radios, and fabrication of the protective aeroshell are on track for a 2028 launch.

Management Notes A NASA Office of Inspector General report noted the mission's launch delay from 2026 to 2028 and a significant cost increase, partly due to supply chain issues and the pandemic.

🏗️ A Global Collaborative Effort

The mission is led by NASA's Johns Hopkins Applied Physics Laboratory (APL) and involves an international team:

Principal Investigator Dr. Elizabeth "Zibi" Turtle.

Key Partners: NASA Goddard, Lockheed Martin, NASA Ames, and others.

International Contributors:

JAXA (Japan)

CNES (France)

DLR (Germany)

In short, Dragonfly is a groundbreaking mission that will use revolutionary technology to explore one of the most Earth-like and chemically rich worlds in our solar system, seeking clues to how life begins.