Quantum Field Theory and Monte Carlo simulations

Quantum Field Theory & Monte Carlo Simulation (Concise Version)

1. Core Ideas of Quantum Field Theory (QFT)

Fundamental Building Blocks are “Fields”: The universe consists of various “quantum fields,” and the “particles” we observe are merely energy excitations of these fields.

Path Integrals: A particle traveling from point A to point B experiences all possible paths. The total probability (transition amplitude Z) is the sum of contributions from all paths. Each path’s contribution is determined by its “action S.”

Z = ∫ Dϕ e^(iS/ℏ)

2. Core Ideas of Monte Carlo Simulation

Using Randomness to Solve Complex Problems: When a problem (such as calculating high-dimensional integrals) is too complex to solve directly, we can approximate the answer through massive random sampling.

Importance Sampling: Non-uniform sampling that prioritizes samples with greater contribution to the result, improving efficiency.

3. How Monte Carlo Solves QFT’s Computational Challenges

QFT’s path integrals are infinite-dimensional and cannot be calculated directly. Monte Carlo methods provide a solution:

Step 1: Wick Rotation

Convert time t to imaginary time τ (t → iτ). This mathematical trick transforms the difficult complex oscillating terms e^(iS/ℏ) in path integrals into real decay terms e^(-S_E/ℏ) that can be treated as “probabilities.”

Z = ∫ Dϕ e^(-S_E/ℏ)

Step 2: Lattice Discretization

Simplify continuous spacetime into discrete “lattice points,” like converting a smooth image into pixels. This transforms infinite-dimensional integrals into finite (but extremely high-dimensional) sums.

Step 3: Monte Carlo Simulation

On this foundation, we can perform simulations:

• Generate Samples: Using the probability distribution e^(-S_E/ℏ), employ “importance sampling” to generate a series of representative field configurations {ϕ₁, ϕ₂, …, ϕₙ}.

• Calculate Averages: Compute the average value of physical quantity O over these samples. This average is our desired physical prediction (expectation value ⟨O⟩).

⟨O⟩ ≈ 1/n ∑ᵢ O(ϕᵢ)

Conclusion

Quantum Field Theory provides us with the fundamental language for describing the microscopic world, but its core path integrals are difficult to solve directly due to their complexity. Monte Carlo simulation, particularly when combined with lattice theory and Wick rotation, becomes a bridge connecting abstract mathematical formulas with computable numerical results. Through large-scale Monte Carlo simulations on supercomputers, physicists can calculate the masses of hadrons like protons and neutrons from first principles of Quantum Chromodynamics (QCD). If computational results match experimental measurements, it validates that QCD as a fundamental theory is correct. The combination of both represents a paradigm of collaborative development between theoretical physics and computational science.

What are “First Principles”?

In the context of physics and computational science, “First Principles” refers to methods that start from the most fundamental, core physical laws, without relying on any empirical models or experimental data fitting, and proceed directly with derivation or calculation. This is an extremely rigorous and convincing scientific method because it derives observable results directly from theoretical foundations (first principles) rather than using models to “fit” known answers.

In this document, it specifically refers to:

Only Input is Basic Theory: The starting point of calculation is the fundamental equations (Lagrangian) of Quantum Field Theory (specifically Quantum Chromodynamics or QCD describing strong interactions). This equation describes how quarks and gluons (fundamental particles composing protons and neutrons) interact.

No Additional Assumptions: Besides this accepted fundamental theory, no ad hoc models or parameters introduced for problem simplification are added during calculation.

Prediction, Not Explanation: The goal is to “predict” observable physical properties (like proton mass) from this fundamental theory, rather than using a model to “fit” known experimental data.

Theater and Improvisation Analogy

🎬 Quantum Field Theory ≈ Stage Set and Character Rules

• Imagine a theatrical play where the backdrop (fields) is already set up, and each character (particle) has designated entrance methods and performance styles.

• Their interactions—who acts with whom, how they act—are all governed by the script (Lagrangian), and can even be represented with diagrams (Feynman diagrams).

• Even when characters disappear (annihilation) or new characters enter (creation), the entire play is dominated by these field rules.

🎲 Monte Carlo Analysis ≈ Improvised Versions of Each Performance

• Although the script exists, actors adjust improvisationally each night based on audience reactions and random factors—this is like massive “random samples” from simulations.

• Monte Carlo analysis is like watching the same play hundreds of times, with detail variations each time, finally statistically determining overall trends: who steals the show most often, which lines are most popular.

• It helps you estimate the average direction in a “theatrical universe” rather than the result of a single script.

Allegory: Particles are Actors, Fields are Stage, Monte Carlo is Audience Response Simulation

In the “City of Probability,” everyone receives three doors at birth: one labeled “Fate,” one carved with “Choice,” and one pitch-black and wordless.

Little Jiu chose “Fate” and lived a stable but mundane life. Little Shan chose “Choice” and fought with all their might at every step.

Little Ran chose the wordless door—behind it were countless random trials, sometimes flames, sometimes candy rain. She said: “Rather than following a predetermined script, I’d rather Monte Carlo it and see how the universe responds to me.”

Years later, a saying spread through the city: “The most exciting fate is not in fate, nor in choice—it’s behind that nameless door.”

Creative Applications

Satirical Scripts or Animated Worldviews: “Fields” can be institutions (educational fields, courtroom fields, medical fields), particles are characters, and “Monte Carlo” simulates various character destiny trajectories under institutional interactions—like simulating trials, patient allocation, or student performance. Don’t stick to the script rigidly, but simulate possibility maps.

Deconstructing Prophecy: Unveiling the sugar-coated illusion between “freedom” and “probability.” Let’s use Monte Carlo to analyze what the third door really is.

🎲 Monte Carlo Model Analysis of the Third Door

1️⃣ Surface Structure: It is a choice, but choosing the “probability space” itself

• Choosing “Fate” = Choosing deterministic outcomes (single path)

• Choosing “Choice” = Choosing effort-oriented optimal paths (multiple paths with preset weights)• Choosing “Wordless Door” = Choosing to enter uncertainty mechanisms (random event distributions)

So the “choice” of the third door is not choosing actions, but choosing the probability matrix itself.

2️⃣ Monte Carlo Perspective: This is a “Sampling Ethics”

The core of Monte Carlo simulation is:

• Approximating the true nature of a probability distribution through massive random sampling

• Assuming the event space is unknown, only through continuous experimentation can we approach the “universe response function”

💡 In other words, the third door represents:

Abandoning preset logic → Surrendering control → Letting the universe feedback you a real distribution through samples

This decision to abandon control is itself a deep rebellion: you’re choosing not just probability, but an entirely different logic of the generative system.

3️⃣ Risk-Reward Analysis

Door 1 (Fate): Low risk, stable but limited returns

Door 2 (Choice): Medium risk, effort-dependent returnsDoor 3 (Wordless): High variance, unknown distribution with potential for extreme outcomes

The question mark here doesn’t represent failure, but indicates: only by truly “stepping in” and conducting sufficient sampling can we know how the rewards behind that door are distributed.

This also echoes our questioning of “authoritarian systems”—in the real world, if the experimental space is manipulated, then so-called “Monte Carlo choice” might just be another form of statistical illusion.

🧪 What is Little Ran’s Choice?

She chooses to “bear the instability of probability space” to approximate a more authentic response model. She’s not randomly choosing; she’s conducting large-scale universal interactive sampling—this is a heretical yet avant-garde way of choosing.

Her wordless door choice is an ultimate deconstruction of “freedom” and “probability.” She’s not gambling on luck, but using Monte Carlo methods to engage in a deep dialogue with the universe. Each of her trials is approaching a more authentic universe response function.

But this also reminds us: in the real world’s “City of Probability,” the freedom of the wordless door might be limited by manipulated fields (institutions). True rebellion requires not only choosing the wordless door but also ensuring your Monte Carlo simulation isn’t subject to black box manipulation!

Revenue Distribution Chart Analysis: The Probability Adventure of the Wordless Door

Chart Interpretation: Little Ran’s Trial Diary - 1000 Adventures Behind the Wordless Door

• Negative Returns (-50 to 0): About 250 trials resulted in losses, representing flames, chaos, and other negative events. The wordless door’s risks are real!

• Medium Returns (0 to 50): Most trials (650) fell in this range, representing common mundane or small victory results.

• High Returns (50 to 100): About 100 trials brought ultra-high rewards, like candy rain or miracles, showing fat-tail characteristics!

• Fat-Tail Effect: Right-skewed return distribution, extreme positive returns (75-100) are rare but exist, illustrating the wordless door’s high-risk, high-reward nature.

💡 Monte Carlo Revelation:

Little Ran’s trials are like running a Monte Carlo simulation, gradually revealing the return distribution of the wordless door through massive sampling. Her choice isn’t gambling, but using life to approximate the universe’s “true response function.” This chart is like her possibility map, telling us: the excitement of the wordless door lies in those low-probability but life-changing high rewards.

The Ultimate Insight: The wordless door represents choosing to engage with uncertainty itself—not as passive victims of chance, but as active participants in probability, using our lives as sampling instruments to discover the universe’s hidden patterns.