# Learner lab record: Field-mode cutoff and observable-difference study

Course: Quantum field theory

Name: ____________________  Date: ____________________  Group: ____________________

## Investigation question

How do cavity geometry, mode cutoff, and subtraction choice affect raw vacuum sums and observable differences?

## Setup

Use the field-mode laboratory. Fix the cavity and field model, increase the mode cutoff systematically, then compare raw sums with a declared difference or renormalized diagnostic.

## Variables

| Variable | Role | Unit |
| --- | --- | --- |
| Cavity length and field mass | physical inputs | length and energy/c² |
| Mode cutoff | regulator | mode count |
| Selected mode frequency | dependent physical spectrum | Hz or energy |
| Raw and subtracted vacuum sums | diagnostics | energy |

## Predict before changing controls

1. Predict the raw zero-point sum as cutoff increases.

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2. Predict what a converged observable difference should do under further cutoff increases.

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## Observation table

| length | mass | cutoff | selected ω | raw sum | subtracted result | cutoff change |
| --- | --- | --- | --- | --- | --- | --- |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |
|   |   |   |   |   |   |   |

## Analyze

1. Which quantity is regulator-dependent?

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2. What physical comparison defines the subtracted observable?

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3. How do boundary conditions select modes?

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4. Why is a divergent formal sum not automatically an extractable reservoir?

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## Evidence-bounded conclusion

Increasing cutoff from ___ to ___ changed the raw sum by ___ while the declared observable ___; therefore the regulator-sensitive quantity is ___.

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