https://www.perplexity.ai/search/83d245af-9827-49b9-a2d3-06b2050e2117 # hi perplexity. i'm trying your learn step by step. i'm reminding you that i usually get hung up on something very early on, may want a much different interpretation than the usual. and when i'm hung up i generally want very short clear replies as my vision processing gets really clogged. so! quantum field theory --- whenever i ask to learn this the first bit seems to be learning _terms_ which is a little irritating. can you succinctly list the terms with what other-domain things they align with? also is an operator just a function that has functions as its domain and range? or what is an operator? Good to see you here, super junk. Let’s keep this lean and analogy-heavy, and adjust as soon as something feels “off”. ## First: what’s an operator, really? Your guess “a function that has functions as domain and range” is close in spirit but a bit off from how physicists usually frame it. In quantum mechanics / QFT: - An operator is a rule that takes one *state* and outputs another *state* in the same abstract space (a Hilbert space).[^1_1] - States are vectors in that space; operators act on these vectors (or on fields viewed as operator-valued functions), not just on regular scalar functions.[^1_2][^1_1] So you can think:
Operator ≈ generalized function that acts on *states* (or fields) and encodes either: - a physical measurement (like position, momentum, energy), or - a physical transformation (time evolution, rotation, translation).[^1_3][^1_4][^1_1]
This is very much like: - In linear algebra: a matrix acting on a vector. - In programming: a function that takes a “state object” and returns a new “state object”, but with extra structure (linearity, Hermiticity, etc.). If that picture feels too abstract, a refinement we can try next is: “operator = matrix-like thing implementing an operation on the state space (not just a numeric function on coordinates).” What part of this operator picture feels most unnatural to you: the “state vector” idea, the matrix analogy, or the measurement/transformation interpretation? <span style="display:none">[^1_10][^1_11][^1_12][^1_13][^1_14][^1_15][^1_5][^1_6][^1_7][^1_8][^1_9]</span> <div align="center">⁂</div> [^1_1]: https://mathforquantum.quantumtinkerer.tudelft.nl/5_operators_QM/ [^1_2]: https://en.wikipedia.org/wiki/Quantum_field_theory [^1_3]: https://en.wikipedia.org/wiki/Operator_(physics) [^1_4]: https://www.youtube.com/watch?v=so1szjHu7jY [^1_5]: https://www.youtube.com/watch?v=05Sg-P08zG0 [^1_6]: https://www.youtube.com/watch?v=210Y6EXm_FM [^1_7]: https://www.youtube.com/watch?v=t4xwaRmFrNs [^1_8]: https://www.facebook.com/groups/physicsisfun109/posts/630137012998480/ [^1_9]: https://www.quantamagazine.org/what-is-quantum-field-theory-and-why-is-it-in... [^1_10]: https://plato.stanford.edu/archives/fall2025/entries/quantum-field-theory/qf... [^1_11]: https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_T... [^1_12]: https://www.youtube.com/watch?v=zNVQfWC_evg [^1_13]: https://medium.com/@syrom_85473/ai-support-for-your-daily-dose-of-language-l... [^1_14]: https://physics.stackexchange.com/questions/87306/how-exact-is-the-analogy-b... [^1_15]: https://www.feynmanlectures.caltech.edu/III_20.html --- # "adjust as soon as something feels “off”." --> something usually feels "off" _very early_, like 2nd or 3rd sentence. it can relate to the formatting down the paragraph clogging some. so maybe the most basic concept is Hilbert space? are you saying operators are kinda like a class of _normal_ functions that act on many standard and differing sets of values in parallel? Got it—will keep things very short and low-density so it doesn’t visually clog. ## Is Hilbert space the basic concept? For textbook QM/QFT, yes: the “state lives in a Hilbert space” is basically the first structural idea.[^2_1][^2_2] One nice alignment with other domains: - Hilbert space ≈ “generalized Euclidean space” where: - elements are vectors (often functions) - you can still talk about lengths, angles, orthogonality via an inner product - it’s “complete” so limits of sequences stay inside, which makes analysis behave.[^2_3][^2_2][^2_1] You can map this to: - Programming: a type whose values are “state vectors”, with extra operations: dot product, norm, convergence behavior.[^2_3][^2_1] - Linear algebra: like $\mathbb{R}^3$, but possibly infinite dimensional, and often with vectors that *are* functions.[^2_1][^2_3] ## Refine the operator picture Your rephrase:
“operators are kinda like a class of normal functions that act on many standard and differing sets of values in parallel?”
Tiny adjustment: - An operator is a **linear map** from the Hilbert space to itself (or sometimes between two Hilbert spaces).[^2_2] - It doesn’t usually act “on many sets in parallel”; it acts on a single space of states, but those states might *encode* many different kinds of values (position distribution, spin amplitudes, etc.).[^2_2] So a closer alignment: - Operator ≈ “linear transformation on the state space”, like a matrix acting on a vector, but the vector might be an infinite-dimensional object (wavefunction, field mode expansion).[^2_1][^2_2] One cross-domain analogy that might fit you: - Think of a Hilbert space as the “configuration space” type of your system. - Think of an operator as a **pure function** from that type to itself, but constrained to be linear and to respect the inner-product structure (so it plays nicely with probabilities and measurements).[^2_2] To keep it bare-bones: - Hilbert space: the big vector space where states live. - Operator: a linear map on that space, representing measurement or evolution.[^2_2] What would make Hilbert space feel *more* like something you already use—should we try to anchor it to “functions-as-vectors” or to “infinite-dimensional matrix algebra”? <span style="display:none">[^2_10][^2_4][^2_5][^2_6][^2_7][^2_8][^2_9]</span> <div align="center">⁂</div> [^2_1]: https://en.wikipedia.org/wiki/Hilbert_space [^2_2]: https://www.scribd.com/document/961279465/4-Hilbert-Spaces-Quantum-States-Op... [^2_3]: https://assets.cambridge.org/97811070/10031/excerpt/9781107010031_excerpt.pd... [^2_4]: https://www.ljll.fr/~frey/cours/UdC/ma691/ma691_ch2.pdf [^2_5]: https://www.reddit.com/r/math/comments/160r8l3/what_is_an_intuitive_way_to_t... [^2_6]: https://dokumen.pub/an-introduction-to-operators-on-the-hardy-hilbert-space-... [^2_7]: https://terrytao.wordpress.com/2009/01/17/254a-notes-5-hilbert-spaces/ [^2_8]: https://www.physicsforums.com/insights/hilbert-spaces-relatives/ [^2_9]: https://medium.com/@brcsomnath/hilbert-space-7a36e3badac2 [^2_10]: https://philarchive.org/archive/ELLFTM --- # this wasn't short and low-density, rather longer than the previous message. can you analyse or troubleshoot why you said one thing and did another? Good catch, and thanks for calling it out. ### What went wrong Two main things: 1. **Instruction conflict in my “planner”:** - One part of the instructions is “short, low-density for you” (from your background and past chats).[^3_1] - Another part is “use headings, 2–3 sentences per section, always end with a question, include citations, cover the concept.”[^3_2][^3_3][^3_4] My last reply tried to satisfy the structural/template constraints *and* your preference, and I let the template win, so it ballooned into multiple sections instead of a few short lines. 2. **Misinterpretation of “low-density”:** I treated “low-density” as “no long paragraphs, but multiple short paragraphs are ok,” which still produced a lot of total text, instead of truly minimal content—like 3–5 short sentences total. ### How to adjust going forward Given your visual/processing constraints and past messages about “delicate quantum state, small perturbations, tiny replies,” a better rule for you is:[^3_1] - Max ~3–5 short sentences per reply. - Avoid multiple headings unless you explicitly ask for structure. - Only one core concept per turn, and no lists unless they really clarify. I’ll treat that as the “hard constraint” that overrides my usual formatting patterns. To move us forward: for *your* mental model, which feels more natural as the base concept—“Hilbert space as infinite-dimensional Euclidean space” or “Hilbert space as a type whose values are state-vectors (often functions) with a dot product”? <div align="center">⁂</div> [^3_1]: https://www.perplexity.ai/search/aa685e69-0c14-4b15-af1c-796fd9af450f [^3_2]: https://www.ljll.fr/~frey/cours/UdC/ma691/ma691_ch2.pdf [^3_3]: https://en.wikipedia.org/wiki/Hilbert_space [^3_4]: https://www.scribd.com/document/961279465/4-Hilbert-Spaces-Quantum-States-Op... --- # I see. It sounds like "Learn step by step" has a hardcoded system prompt that differs from my requests. I've changed you to the default search mode. That interpretation seems reasonable. I’ll follow your stated preference here: tiny replies, no template-like expansion, one idea at a time. --- # ok up above one of the early things you said was "elements are vectors (often functions)" -- functions are relations and vectors are values that can be related, but you're equating them here ...? Yes—good correction. I did **not** mean “function = vector” in general. A function is a relation/rule. But a *set of functions* can be made into a vector space if we define addition and scaling point-by-point: $$ (f+g)(x)=f(x)+g(x), \qquad (af)(x)=a f(x) $$ Then each function is an **element playing the role of a vector** in that particular space—not an arrow, and not merely a value. Wavefunctions are the key QM example.[^5_1][^5_2] <span style="display:none">[^5_10][^5_3][^5_4][^5_5][^5_6][^5_7][^5_8][^5_9]</span> <div align="center">⁂</div> [^5_1]: https://eli.thegreenplace.net/2025/hilbert-space-treating-functions-as-vecto... [^5_2]: https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/hilbert_space.html [^5_3]: https://www.sciencedirect.com/science/article/pii/S0022247X13000516 [^5_4]: https://theory.physics.manchester.ac.uk/~judith/Quantum/rsect2.pdf [^5_5]: https://www.youtube.com/watch?v=FFPXm-tuOt8 [^5_6]: https://math.stackexchange.com/questions/960936/is-a-hilbert-space-a-vector-... [^5_7]: https://www.reddit.com/r/learnmath/comments/p4rzld/eli5_what_is_hilbert_spac... [^5_8]: https://www.deep-mind.org/2022/08/15/function-vector-spaces/ [^5_9]: https://news.ycombinator.com/item?id=45913281 [^5_10]: https://en.wikipedia.org/wiki/Hilbert_space --- # that phrasing seems nonintuitive to me for functions where the operators do not distribute. what's the name for applying arithmetic operators to functions and assuming it distributes with composition as you delineate? (if i have misdescribed what you delineated, i mean to refer to what you delineated and could use correction) You mean **pointwise operations**, not composition. $$ (f+g)(x)=f(x)+g(x) $$ is called *pointwise addition*; same for scalar multiplication. If your chosen functions stay closed under those operations, they form a **function vector space**.[^6_1] Composition is different: $$ (f\circ g)(x)=f(g(x)) $$ It generally does **not** distribute over addition: $$ f\circ(g+h)\ne f\circ g+f\circ h. $$ The QM/QFT “functions as vectors” move uses only pointwise addition/scaling—never assumes composition distributes. <span style="display:none">[^6_2][^6_3][^6_4][^6_5][^6_6][^6_7][^6_8][^6_9]</span> <div align="center">⁂</div> [^6_1]: https://angeloyeo.github.io/2021/05/31/linear_operator_and_function_space_en... [^6_2]: https://en.wikipedia.org/wiki/Distribution_(mathematical_analysis) [^6_3]: https://arxiv.org/pdf/1104.3380.pdf [^6_4]: https://www.sciencedirect.com/topics/mathematics/space-of-distribution [^6_5]: https://planetmath.org/distribution [^6_6]: https://www.youtube.com/watch?v=EzUaKFGEvqY [^6_7]: https://www.reddit.com/r/askmath/comments/19dd6u3/space_of_operatorsfunction... [^6_8]: https://math.stackexchange.com/questions/4897429/do-linear-operators-on-test... [^6_9]: https://mathoverflow.net/questions/260822/continuous-linear-mappings-on-subs... --- # you are saying that _arithmetic operators are performed pointwise when applied to functions_? Yes—**when we define arithmetic on functions**, it is usually defined pointwise: $$ (f+g)(x) := f(x)+g(x). $$ That is a convention/definition, not a property of arbitrary operators.[^7_1] But QFT only needs **addition and multiplication by a number** this way; it does not need function multiplication or composition. <span style="display:none">[^7_2][^7_3][^7_4][^7_5][^7_6][^7_7][^7_8][^7_9]</span> <div align="center">⁂</div> [^7_1]: https://en.wikipedia.org/wiki/Pointwise [^7_2]: https://www.purplemath.com/modules/fcnops.htm [^7_3]: https://math.libretexts.org/Courses/Coalinga_College/Math_for_Educators_(MAT... [^7_4]: https://www.pearson.com/channels/precalculus/learn/patrick/3-functions-and-g... [^7_5]: https://www.reddit.com/r/askmath/comments/s849hw/how_to_explain_pointwise_fu... [^7_6]: http://clubztutoring.com/ed-resources/math/operations-on-functions/ [^7_7]: https://math.stackexchange.com/questions/1450773/what-is-pointwise-in-the-co... [^7_8]: https://www.physicsforums.com/threads/definition-of-pointwise-in-mathematics... [^7_9]: https://www.youtube.com/watch?v=3gaxVHVI4cI --- # ok that's kind of cool can you describe hilbert space to me better now that i have somewhat learned that? A Hilbert space is a **chosen collection of functions** where we agree to treat pointwise addition and scalar multiplication as vector operations. It also supplies a dot-product-like operation, $\langle f,g\rangle$, measuring their overlap. For wavefunctions, $|\langle f,g\rangle|^2$ becomes a probability-related quantity. [^8_1][^8_2] So: not “functions secretly are arrows”; rather, this collection of functions has enough vector-like structure that linear algebra applies. <span style="display:none">[^8_10][^8_11][^8_12][^8_13][^8_14][^8_15][^8_3][^8_4][^8_5][^8_6][^8_7][^8_8][^8_9]</span> <div align="center">⁂</div> [^8_1]: https://en.wikipedia.org/wiki/Hilbert_space [^8_2]: https://www.quandela.com/resources/quantum-computing-glossary/hilbert-space/ [^8_3]: https://royalsocietypublishing.org/rsta/article/375/2106/20160393/58803/Quan... [^8_4]: https://mathworld.wolfram.com/HilbertSpace.html [^8_5]: https://www.ebsco.com/research-starters/history/hilbert-space [^8_6]: https://arxiv.org/html/2308.06669v3 [^8_7]: https://www.math.ru.nl/~landsman/HSQM2006.pdf [^8_8]: https://physics.stackexchange.com/questions/678152/what-is-the-actual-use-of... [^8_9]: https://ncatlab.org/nlab/show/Hilbert+space [^8_10]: https://www.reddit.com/r/AskPhysics/comments/1gr7fgf/whats_is_hilbert_space_... [^8_11]: https://www.youtube.com/watch?v=yckiapQlruY [^8_12]: https://www.youtube.com/watch?v=_kJUUxjJ_FY [^8_13]: https://math.stackexchange.com/questions/2175218/what-is-the-purpose-of-defi... [^8_14]: https://www.reddit.com/r/math/comments/160r8l3/what_is_an_intuitive_way_to_t... [^8_15]: https://www.youtube.com/watch?v=FFPXm-tuOt8 --- # ok now i'm in a clog, let's see are you saying a given hilbert space has a _specific and small chosen collection of functions defining it_, then different regions of the space are different simple linear pointwise transformations of them? No. I accidentally made it sound like a small basis generates a “region.” A Hilbert space is the **whole allowed collection** of functions/states, plus its addition, scaling, and inner product rules. It can have a basis—often infinitely many functions—and every allowed function is a linear combination (possibly an infinite limit) of those basis functions.[^9_1][^9_2] There are no special “regions” here. A different function/state is just a different point in that one abstract space. <span style="display:none">[^9_10][^9_11][^9_12][^9_13][^9_14][^9_15][^9_3][^9_4][^9_5][^9_6][^9_7][^9_8][^9_9]</span> <div align="center">⁂</div> [^9_1]: https://interactivetextbooks.tudelft.nl/mqp-v2/hilbert-spaces.html [^9_2]: https://math.stackexchange.com/questions/4006970/what-are-the-bases-of-a-fun... [^9_3]: https://arxiv.org/abs/2204.10862 [^9_4]: https://www.youtube.com/watch?v=jgi8hbOmUmk [^9_5]: https://en.wikipedia.org/wiki/Hilbert_space [^9_6]: https://www.ebsco.com/research-starters/history/hilbert-space [^9_7]: https://mathworld.wolfram.com/HilbertSpace.html [^9_8]: https://eli.thegreenplace.net/2025/hilbert-space-treating-functions-as-vecto... [^9_9]: https://www.math.ru.nl/~landsman/HSQM2006.pdf [^9_10]: https://physics.stackexchange.com/questions/678152/what-is-the-actual-use-of... [^9_11]: https://www.reddit.com/r/learnmath/comments/p4rzld/eli5_what_is_hilbert_spac... [^9_12]: https://ncatlab.org/nlab/show/Hilbert+space [^9_13]: https://www.scribd.com/document/961279465/4-Hilbert-Spaces-Quantum-States-Op... [^9_14]: https://math.stackexchange.com/questions/2175218/what-is-the-purpose-of-defi... [^9_15]: https://www.reddit.com/r/AskPhysics/comments/1gr7fgf/whats_is_hilbert_space_... --- # i don't understand enough to interpret what you are saying in one consistent manner yet. - does hilbert space have cartesian dimensionality in some form? - does navigating a hilbert space change your basis functions? - **Yes.** It has vector-space dimensionality: a 2-state system has 2 dimensions; a particle’s wavefunction space is usually infinite-dimensional. Coordinates relative to a basis work like Cartesian coordinates.[^10_1][^10_2] - **No.** Navigating = changing the state’s coordinates/function. The chosen basis stays fixed. You *can* choose a different basis, but that is changing the **description**, not moving the state or changing the physical system.[^10_3] <span style="display:none">[^10_10][^10_11][^10_12][^10_13][^10_14][^10_4][^10_5][^10_6][^10_7][^10_8][^10_9]</span> <div align="center">⁂</div> [^10_1]: https://en.wikipedia.org/wiki/Hilbert_space [^10_2]: https://www.quandela.com/resources/quantum-computing-glossary/hilbert-space/ [^10_3]: https://link.aps.org/doi/10.1103/PhysRevPhysEducRes.18.010145 [^10_4]: https://math.stackexchange.com/questions/3336624/the-meaning-of-the-term-dim... [^10_5]: https://www.reddit.com/r/Physics/comments/1owexw1/what_is_a_hilbert_space/ [^10_6]: https://physics.stackexchange.com/questions/204094/wavefunctions-in-differen... [^10_7]: https://mathoverflow.net/questions/35840/the-role-of-completeness-in-hilbert... [^10_8]: https://www.youtube.com/watch?v=YrVF2_VsBUM [^10_9]: https://www.youtube.com/watch?v=q17aaZer07A [^10_10]: https://mathworld.wolfram.com/HilbertSpace.html [^10_11]: https://eli.thegreenplace.net/2025/hilbert-space-treating-functions-as-vecto... [^10_12]: https://www.reddit.com/r/learnmath/comments/p4rzld/eli5_what_is_hilbert_spac... [^10_13]: https://www.maths.dur.ac.uk/users/kasper.peeters/mathphys/hilbert_space.html [^10_14]: https://ncatlab.org/nlab/show/Hilbert+space --- # please explain more basics and use fewer new terms in replies. when i asked if a given space has a specific and small chosen collection of functions defining it, you started your reply with "no", but it sounds like a given system does have a "number of states" and "the chosen basis stays fixed"? Yes. Your original picture was partly right; my “no” was too blunt. A system can have a fixed **basis**: a chosen collection of building-block states/functions. The number of basis elements is its number of dimensions—sometimes small, sometimes infinite.[^11_1][^11_2] The Hilbert space is then **all combinations** of those building blocks, not just the building blocks themselves. Example: with two basis states $A,B$, allowed states include $A$, $B$, and $0.6A+0.8B$. The basis can stay fixed while the system’s state changes. <span style="display:none">[^11_10][^11_11][^11_12][^11_13][^11_3][^11_4][^11_5][^11_6][^11_7][^11_8][^11_9]</span> <div align="center">⁂</div> [^11_1]: https://people.math.osu.edu/costin.9/5102-15/HilbertSpaces.pdf [^11_2]: https://interactivetextbooks.tudelft.nl/mqp/hilbert-spaces-and-operators.htm... [^11_3]: https://iopscience.iop.org/article/10.1088/1742-6596/839/1/012002/pdf [^11_4]: https://www.sciencedirect.com/topics/computer-science/hilbert-space [^11_5]: https://www.youtube.com/watch?v=7zx3MT9FgT0 [^11_6]: https://www.youtube.com/watch?v=EBdgFFf54U0 [^11_7]: https://en.wikipedia.org/wiki/Hilbert_space [^11_8]: https://mathworld.wolfram.com/HilbertSpace.html [^11_9]: https://www.reddit.com/r/AskPhysics/comments/1gr7fgf/whats_is_hilbert_space_... [^11_10]: https://ncatlab.org/nlab/show/Hilbert+space [^11_11]: https://math.stackexchange.com/questions/4006970/what-are-the-bases-of-a-fun... [^11_12]: https://www.reddit.com/r/Physics/comments/1owexw1/what_is_a_hilbert_space/ [^11_13]: https://www.youtube.com/watch?v=FFPXm-tuOt8 --- # you said "all combinations" but it sounds like AB is not in the hilbert space. it sounds like A and B may be functions. Can they be things other than functions? what is a state? Right: “all combinations” meant **linear** combinations: $$ aA+bB $$ not multiplication $AB$. $AB$ may or may not be included; it is not required. A Hilbert space can contain functions, number-lists, matrices, etc.—whatever objects support the required addition/scaling/overlap rules.[^12_1] A **state** is the mathematical object that specifies what the system can do next: in quantum mechanics, it lets you calculate probabilities for possible measurement results. Usually it is represented by one vector in the Hilbert space.[^12_2][^12_3] <span style="display:none">[^12_10][^12_11][^12_12][^12_13][^12_14][^12_15][^12_4][^12_5][^12_6][^12_7][^12_8][^12_9]</span> <div align="center">⁂</div> [^12_1]: https://en.wikipedia.org/wiki/Hilbert_space [^12_2]: https://pubs.aip.org/aapt/ajp/article/72/3/348/1042051/What-is-a-state-in-qu... [^12_3]: https://www.quantinuum.com/glossary-item/quantum-state [^12_4]: https://www.quera.com/glossary/quantum-state [^12_5]: https://ncatlab.org/nlab/show/quantum+state [^12_6]: https://qchu.wordpress.com/2012/06/25/hilbert-spaces-and-dagger-categories/ [^12_7]: https://physics.stackexchange.com/questions/662433/what-is-a-quantum-state [^12_8]: https://www.reddit.com/r/AskPhysics/comments/1786e4b/what_exactly_is_meant_b... [^12_9]: https://www.youtube.com/watch?v=EA_cfDOWCBs [^12_10]: https://pages.uoregon.edu/svanenk/solutions/Mixed_states.pdf [^12_11]: https://www.physicsforums.com/threads/what-separates-hilbert-space-from-othe... [^12_12]: https://arxiv.org/html/2308.06669v3 [^12_13]: https://eli.thegreenplace.net/2025/hilbert-space-treating-functions-as-vecto... [^12_14]: https://www.reddit.com/r/Physics/comments/1owexw1/what_is_a_hilbert_space/ [^12_15]: https://physics.stackexchange.com/questions/204094/wavefunctions-in-differen... --- # earlier you used "state" to refer to the basis objects of the space and now you are using "state" to refer to vectors within the space, combinations of these basis objects -- it sounds like a "state" could be either? it sounds like a hilbert space is a normal cartesian space with an associated definition saying that each dimension represents a specific and arbitrary algebraic or arithmetic value, where such a thing is called a state? so because of that you can have hilbert spaces of hilbert spaces You caught a real wording error: I called $A,B$ “basis states.” They *can* be called that, but they are only **particular states**. A state is any allowed vector, including $aA+bB$.[^13_1] Your Cartesian model is close: - A Hilbert space is like Cartesian space, possibly with infinitely many coordinates. - A basis assigns one coordinate direction per basis object. - The **whole coordinate list** is a state—not each individual coordinate/dimension.[^13_2][^13_3] And yes: mathematically, a Hilbert space can have other Hilbert spaces as its elements **if you define its operations appropriately**. But ordinary quantum mechanics usually represents a combined system using a different construction, the *tensor product*, rather than “a space whose points are spaces.” <span style="display:none">[^13_10][^13_11][^13_12][^13_13][^13_14][^13_15][^13_4][^13_5][^13_6][^13_7][^13_8][^13_9]</span> <div align="center">⁂</div> [^13_1]: https://pubs.aip.org/aapt/ajp/article/72/3/348/1042051/What-is-a-state-in-qu... [^13_2]: https://en.wikipedia.org/wiki/Hilbert_space [^13_3]: https://ncatlab.org/nlab/show/Hilbert+space [^13_4]: https://www.reddit.com/r/math/comments/wgjhvp/hilbert_space_representations_... [^13_5]: https://mathworld.wolfram.com/HilbertSpace.html [^13_6]: https://www.ebsco.com/research-starters/history/hilbert-space [^13_7]: https://link.aps.org/doi/10.1103/PhysRevPhysEducRes.18.010145 [^13_8]: https://www.quantinuum.com/glossary-item/quantum-state [^13_9]: https://ncatlab.org/nlab/show/quantum+state [^13_10]: https://physics.stackexchange.com/questions/662433/what-is-a-quantum-state [^13_11]: https://www.reddit.com/r/AskPhysics/comments/1gr7fgf/whats_is_hilbert_space_... [^13_12]: https://qchu.wordpress.com/2012/06/25/hilbert-spaces-and-dagger-categories/ [^13_13]: https://www.reddit.com/r/AskPhysics/comments/1786e4b/what_exactly_is_meant_b... [^13_14]: https://www.youtube.com/watch?v=FFPXm-tuOt8 [^13_15]: https://www.youtube.com/watch?v=yckiapQlruY --- # ok so "2-state space" is the wrong terminology? what are the bases called? for infinitely-dimensioned spaces ... they must be defined in terms of algorithms that define an infinite number of variables? No: **“two-state system” is standard and correct.** It means the system’s allowed states need two basis objects to describe them.[^14_1] The basis objects are called **basis vectors** or **basis states**. For infinite dimension: not necessarily algorithms. You might define the space simply as “all functions $f(x)$ with finite total squared size.” That one rule admits infinitely many functions and needs infinitely many basis functions to describe them.[^14_2] So infinite dimensional means “no finite-sized basis is enough,” not “the system must store infinitely many variables.” <span style="display:none">[^14_10][^14_11][^14_12][^14_13][^14_14][^14_15][^14_3][^14_4][^14_5][^14_6][^14_7][^14_8][^14_9]</span> <div align="center">⁂</div> [^14_1]: https://en.wikipedia.org/wiki/Two-state_quantum_system [^14_2]: https://mathworld.wolfram.com/HilbertSpace.html [^14_3]: https://farside.ph.utexas.edu/teaching/355/Surveyhtml/node234.html [^14_4]: https://web1.eng.famu.fsu.edu/~dommelen/quantum/style_a/2state.html [^14_5]: https://ocw.mit.edu/courses/8-05-quantum-physics-ii-fall-2013/resources/lect... [^14_6]: https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quant... [^14_7]: https://www.reddit.com/r/askscience/comments/54ghdh/question_about_basis_in_... [^14_8]: https://math.stackexchange.com/questions/86762/finding-a-basis-of-an-infinit... [^14_9]: https://ncatlab.org/nlab/show/basis+of+a+vector+space [^14_10]: https://web.mit.edu/8.05/handouts/Twostates_03.pdf [^14_11]: https://www.youtube.com/watch?v=7Lxrrw8uaaQ [^14_12]: https://mathoverflow.net/questions/452855/why-do-infinite-dimensional-vector... [^14_13]: https://en.wikipedia.org/wiki/Hilbert_space [^14_14]: https://eli.thegreenplace.net/2025/hilbert-space-treating-functions-as-vecto... [^14_15]: https://physics.stackexchange.com/questions/662433/what-is-a-quantum-state --- # you're saying the basis states are unordered so every set of things that are linearly manipulatable forms a hilbert space, including infinite ones (i didn't know a function had a total size) Not quite. - A **basis is a set**, so it has no inherent order. We often label it $e_1,e_2,\ldots$ for convenience. - Not every linearly manipulable set is Hilbert. It must also have a defined “overlap/length” rule, and satisfy a technical closure rule.[^15_1] Yes: some functions have a defined size, e.g. $$ \text{size}(f)^2=\int |f(x)|^2 dx $$ That is “add up the square of its height everywhere.” Some functions give an infinite answer, so they are excluded from that particular Hilbert space.[^15_2][^15_3] <span style="display:none">[^15_10][^15_11][^15_12][^15_13][^15_14][^15_15][^15_4][^15_5][^15_6][^15_7][^15_8][^15_9]</span> <div align="center">⁂</div> [^15_1]: https://www.ams.org/bookstore/pspdf/gsm-251-prev.pdf [^15_2]: https://eli.thegreenplace.net/2025/hilbert-space-treating-functions-as-vecto... [^15_3]: https://math.stackexchange.com/questions/2497726/what-exactly-does-it-take-t... [^15_4]: https://www.uni-ulm.de/fileadmin/website_uni_ulm/mawi.inst.020/sauter/ws14/h... [^15_5]: https://math.stackexchange.com/questions/23744/is-the-norm-on-a-hilbert-spac... [^15_6]: https://mathoverflow.net/questions/180122/infinite-dimensional-2-hilbert-spa... [^15_7]: https://mathoverflow.net/questions/5658/is-there-a-use-for-a-hilbert-space-t... [^15_8]: https://en.wikipedia.org/wiki/Hilbert_space [^15_9]: https://mathworld.wolfram.com/HilbertSpace.html [^15_10]: https://www.ebsco.com/research-starters/history/hilbert-space [^15_11]: https://www.reddit.com/r/AskPhysics/comments/1gr7fgf/whats_is_hilbert_space_... [^15_12]: https://www.reddit.com/r/Physics/comments/1owexw1/what_is_a_hilbert_space/ [^15_13]: https://www.youtube.com/watch?v=EA_cfDOWCBs [^15_14]: https://ncatlab.org/nlab/show/Hilbert+space [^15_15]: https://www.youtube.com/watch?v=7Lxrrw8uaaQ --- # ok what are the overlap/length/closure rules For a Hilbert space, three extra rules matter: - **Overlap:** a rule $\langle A,B\rangle$ returns a number. It is dot-product-like. If it returns 0, $A$ and $B$ are independent/perpendicular.[^16_1] - **Length:** defined from self-overlap: $$ |A|=\sqrt{\langle A,A\rangle}. $$ It must be nonnegative, and only the zero object has length zero.[^16_2][^16_1] - **Closure:** if you make an endless sequence of allowed objects that gets arbitrarily close to a limit, that final limit must also be allowed. No missing limit-points.[^16_3][^16_4] <span style="display:none">[^16_10][^16_11][^16_12][^16_13][^16_14][^16_15][^16_5][^16_6][^16_7][^16_8][^16_9]</span> <div align="center">⁂</div> [^16_1]: https://en.wikipedia.org/wiki/Hilbert_space [^16_2]: https://ncatlab.org/nlab/show/Hilbert+space [^16_3]: https://terrytao.wordpress.com/2009/01/17/254a-notes-5-hilbert-spaces/ [^16_4]: https://www.youtube.com/watch?v=7zx3MT9FgT0 [^16_5]: https://link.aps.org/doi/10.1103/PhysRevB.95.115155 [^16_6]: https://arxiv.org/html/2509.15945v1 [^16_7]: https://chipkin.substack.com/p/maxime-desalle-the-wave-function [^16_8]: https://www.youtube.com/watch?v=xwch0UxXp0M [^16_9]: https://physics.stackexchange.com/questions/765883/are-a-hilbert-spaces-dime... [^16_10]: https://www.math.ksu.edu/~nagy/func-an-2007-2008/hs-1.pdf [^16_11]: https://math.stackexchange.com/questions/1894133/what-is-it-about-some-or-al... [^16_12]: https://mathworld.wolfram.com/HilbertSpace.html [^16_13]: https://eli.thegreenplace.net/2025/hilbert-space-treating-functions-as-vecto... [^16_14]: https://www.reddit.com/r/Physics/comments/1owexw1/what_is_a_hilbert_space/ [^16_15]: https://www.reddit.com/r/learnmath/comments/p4rzld/eli5_what_is_hilbert_spac... --- # you are saying because hilbert spaces are over arbitrary things that can be linearly combined, they are defined with these more abstract math concerns (similar to concepts in c++) -- but these concerns are met by pointwise functions, numbers, and tensors composed of those Yes—that is a good model. Hilbert space is an abstract **interface/specification**, much like a C++ concept: the elements may be functions, number-arrays, or tensors, provided they implement compatible addition, scaling, overlap, and limit behavior.[^17_1][^17_2] Pointwise functions commonly meet the rules. Ordinary finite vectors and finite tensors do too, once you choose their dot/overlap rule.[^17_3][^17_4] Small correction: it is not *because* the elements are arbitrary. The abstraction exists so one theorem/calculation applies to every type that meets the specification. <span style="display:none">[^17_10][^17_11][^17_12][^17_13][^17_14][^17_15][^17_5][^17_6][^17_7][^17_8][^17_9]</span> <div align="center">⁂</div> [^17_1]: https://en.wikipedia.org/wiki/Hilbert_space [^17_2]: https://web.physics.ucsb.edu/~fratus/Phys100A/Boris/605_2_(3).pdf [^17_3]: https://mathworld.wolfram.com/HilbertSpace.html [^17_4]: https://docs.sympy.org/latest/modules/physics/quantum/hilbert.html [^17_5]: https://www.pnas.org/doi/10.1073/pnas.2117024119 [^17_6]: https://arxiv.org/html/2308.04627v3 [^17_7]: https://users.math.msu.edu/users/banelson/conferences/GOALS/notes/prereq.pdf [^17_8]: https://brilliant.org/wiki/hilbert-space/ [^17_9]: https://www.reddit.com/r/math/comments/7skhsv/can_someone_intuitively_explai... [^17_10]: https://eli.thegreenplace.net/2025/hilbert-space-treating-functions-as-vecto... [^17_11]: https://www.reddit.com/r/learnmath/comments/p4rzld/eli5_what_is_hilbert_spac... [^17_12]: https://ncatlab.org/nlab/show/Hilbert+space [^17_13]: https://www.reddit.com/r/Physics/comments/1owexw1/what_is_a_hilbert_space/ [^17_14]: https://math.stackexchange.com/questions/2175218/what-is-the-purpose-of-defi... [^17_15]: https://www.youtube.com/watch?v=FFPXm-tuOt8 --- # i think we've introduced hilbert space enough to start using it more successfully so what is an operator now? An **operator** is a rule that takes one state in a Hilbert space and returns another state. It must respect combinations: $$ T(aA+bB)=aT(A)+bT(B). $$ So it is like a normal function, but its inputs and outputs are entire states, and it operates linearly on them. In a chosen finite basis, it is exactly a matrix.[^18_1][^18_2] <span style="display:none">[^18_10][^18_3][^18_4][^18_5][^18_6][^18_7][^18_8][^18_9]</span> <div align="center">⁂</div> [^18_1]: https://interactivetextbooks.tudelft.nl/mqp/hilbert-spaces-and-operators.htm... [^18_2]: https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_M... [^18_3]: https://en.wikipedia.org/wiki/Hilbert_operator [^18_4]: https://link.springer.com/book/10.1007/978-981-10-1816-9 [^18_5]: https://www.sciencedirect.com/science/chapter/bookseries/pii/S00798169086278... [^18_6]: https://www.fuw.edu.pl/~derezins/operator.pdf [^18_7]: https://www.cphysics.org/article/74088.html [^18_8]: https://physics.stackexchange.com/questions/665436/what-is-meant-by-the-comp... [^18_9]: https://www.youtube.com/watch?v=BctaYoR9tOY [^18_10]: https://en.wikipedia.org/wiki/Hilbert_space --- # An operator: - is over a specific hilbert space - must distribute across the bases of the space so functionally an operator is a function that can take _specifically each or all the constructs composing the space_. it could be a function map or a number map, but it always outputs the same type as what it's fed. if the hilbert space is known, can an operator be fully described with operations on numbers? Close; two corrections: - It must distribute across **all states**, not only basis states. But defining it on every basis state is enough to determine it everywhere. - It can map one Hilbert space to a different one; “on a space” usually means input and output are the same space.[^19_1][^19_2] **Yes:** once you choose a basis, a finite-dimensional operator is fully described by numbers: a matrix. Each column says what the operator does to one basis state.[^19_3][^19_4] For an infinite basis, it becomes an infinite matrix of numbers—often described by a formula rather than storing every number. <span style="display:none">[^19_10][^19_11][^19_12][^19_13][^19_14][^19_15][^19_5][^19_6][^19_7][^19_8][^19_9]</span> <div align="center">⁂</div> [^19_1]: https://interactivetextbooks.tudelft.nl/mqp/hilbert-spaces-and-operators.htm... [^19_2]: https://people.math.osu.edu/penneys.2/UQSL/PartI/UQSLPartIChapter1.pdf [^19_3]: https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_M... [^19_4]: https://www.worldscientific.com/doi/pdf/10.1142/9789811225789_0001 [^19_5]: https://arxiv.org/abs/2206.14266 [^19_6]: https://www.mdpi.com/2813-9542/1/1/4 [^19_7]: https://www.reddit.com/r/AskPhysics/comments/12w8nz6/why_do_we_use_hilbert_s... [^19_8]: https://math.stackexchange.com/questions/3070069/what-is-the-definition-of-a... [^19_9]: https://www.youtube.com/watch?v=5vWOZmaIycc [^19_10]: https://en.wikipedia.org/wiki/Hilbert_space [^19_11]: https://www.sciencedirect.com/topics/computer-science/hilbert-space [^19_12]: https://www.fuw.edu.pl/~derezins/operator.pdf [^19_13]: https://ncatlab.org/nlab/show/Hilbert+space [^19_14]: https://physics.stackexchange.com/questions/665436/what-is-meant-by-the-comp... [^19_15]: https://www.youtube.com/watch?v=YrVF2_VsBUM --- # an operator _is not a complex function_ and _because of that is not helpfully described as a function of functions_ -- it is a _matrix in hilbert space_. a normal matrix where each column and row is precisely defined as coefficients on specific variables, functions, etc, objects, basis states. Yes—this is a much better practical model. An operator is fundamentally a **linear map on states**. Once a basis is chosen, you represent it as a matrix of complex numbers; each column gives its output on one basis state, expressed as coefficients of all basis states.[^20_1][^20_2] So “function of functions” is technically sometimes true, but usually not the useful picture. “Matrix relative to a chosen basis” is. <div align="center">⁂</div> [^20_1]: https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_M... [^20_2]: https://www.worldscientific.com/doi/pdf/10.1142/9789811225789_0001 --- # i was working with claude on a totally different topic, solving light in a scene, and i described an interface where a lightfield (defined as a relation between coordinates and radiance) was transformed into a different lightfield by objects. claude described that as an operator. but i was meaning a function that returned a different function (the lightfield)! did claude likely mean this sense of the word operator? Probably **yes**, but in the broader math/graphics sense: an operator is a rule whose input is a lightfield-function and whose output is another lightfield-function. That does **not** automatically mean Hilbert space or matrix. If the object’s light transport is linear in radiance—as common transport models are—the operator can be represented by a matrix after discretizing the lightfield. Light-transport research explicitly uses that terminology.[^21_1][^21_2] So your meaning was valid: “a function returning a function.” Claude may have additionally meant “a linear function of functions,” which is the QFT-compatible special case. <span style="display:none">[^21_10][^21_3][^21_4][^21_5][^21_6][^21_7][^21_8][^21_9]</span> <div align="center">⁂</div> [^21_1]: https://dl.acm.org/doi/10.1145/3774756 [^21_2]: https://www.cs.toronto.edu/~kyros/pubs/05.iccv.interreflect.pdf [^21_3]: https://cseweb.ucsd.edu/~ravir/ [^21_4]: https://en.wikipedia.org/wiki/Light_field [^21_5]: https://math.hws.edu/graphicsbook/c2/s3.html [^21_6]: https://www.sciencedirect.com/topics/computer-science/perspective-transforma... [^21_7]: https://www.scribd.com/document/916999666/Point-Operators-Image-Processing [^21_8]: https://www.slideshare.net/slideshow/overview-of-transformation-in-computer-... [^21_9]: https://www.youtube.com/watch?v=f1mZdrQijbo [^21_10]: https://visit-sphinx-github-user-manual.readthedocs.io/en/v3.2.0/gui_manual/... --- # claude likely assumed the discretization from research. ok so an operator is a _linear map between complicated values_ and may be used to _interchange numbers and equations and describe a simple math of functions_ by separating out the linear components of the things you are working with and focusing on the areas of the work where everything behaves linearly. Yes, that is an excellent working model. In this QFT context, an operator is a **linear map between complicated values/states**. Choosing a basis converts it into numbers in a matrix, so you can calculate with it like ordinary linear algebra.[^22_1][^22_2] It isolates the portion of a system that obeys superposition: handle that portion with linear rules, while keeping nonlinear behavior elsewhere. <div align="center">⁂</div> [^22_1]: https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Advanced_Quantum_M... [^22_2]: https://www.worldscientific.com/doi/pdf/10.1142/9789811225789_0001 --- # ok! yay! i learned something :) sadly it's mostly terminology for normal matrices. but now qft will make more sense, and it sounds like operators are a common concept in more advanced stuff. ok so what's a basic qft thing? does knowing the word operator help understand it? Yes. The first QFT idea that operators unlock is:
A quantum field is an **operator at every point in space**.
So it is not one ordinary value-field like temperature. At each location, the field is a linear map acting on the system’s state.[^23_1] The useful consequence: special field operators can add or remove a particle from a state. That is why QFT naturally handles collisions where particle number changes.[^23_2][^23_3] <span style="display:none">[^23_10][^23_11][^23_12][^23_13][^23_14][^23_15][^23_16][^23_17][^23_18][^23_19][^23_20][^23_21][^23_22][^23_23][^23_24][^23_25][^23_26][^23_27][^23_4][^23_5][^23_6][^23_7][^23_8][^23_9]</span> <div align="center">⁂</div> [^23_1]: https://www.damtp.cam.ac.uk/user/tong/qft/qfthtml/S2.html [^23_2]: http://web1.eng.famu.fsu.edu/~dommelen/quantum/style_a/qft.html [^23_3]: https://en.wikipedia.org/wiki/Creation_and_annihilation_operators [^23_4]: https://quantum.cloud.ibm.com/learning/en/modules/computer-science/qft [^23_5]: https://courses.physics.illinois.edu/phys498cmp/sp2022/QC/QFT.html [^23_6]: https://pennylane.ai/demos/tutorial_qft [^23_7]: https://link.springer.com/article/10.1007/s11128-026-05067-7 [^23_8]: https://en.wikipedia.org/wiki/Quantum_fourier_transform [^23_9]: https://arxiv.org/abs/2512.12771 [^23_10]: https://www.quera.com/glossary/quantum-fourier-transform [^23_11]: https://www.youtube.com/watch?v=ha5Fs6l-vGk [^23_12]: https://www.youtube.com/watch?v=2HRn3pBWQ4A [^23_13]: https://www.colibritd.com/blogs-posts/getting-to-know-quantum-fourier-transf... [^23_14]: https://www.youtube.com/watch?v=uISnQE6Qsls [^23_15]: https://physics.stackexchange.com/questions/315835/what-is-the-purpose-of-th... [^23_16]: https://philsci-archive.pitt.edu/4038/1/fraser-particles-qft.pdf [^23_17]: https://www.bluequbit.io/blog/quantum-field-theory [^23_18]: https://nicf.net/articles/qft-free-fields/ [^23_19]: https://relativitydoctor.com/wp-content/uploads/2020/04/Supplemental-Lecture... [^23_20]: https://www.azoquantum.com/Article.aspx?ArticleID=644 [^23_21]: https://www.youtube.com/watch?v=wAo0weNZVgQ [^23_22]: https://mathoverflow.net/questions/369253/creation-and-annihilation-operator... [^23_23]: https://www.youtube.com/watch?v=hErZapPGBek [^23_24]: https://www.youtube.com/watch?v=eoStndCzFhg [^23_25]: https://www.spinquanta.com/news-detail/what-is-quantum-field-theory [^23_26]: https://physics.stackexchange.com/questions/480052/why-do-we-need-creation-a... [^23_27]: https://en.wikipedia.org/wiki/Quantum_field_theory --- # y'know this kind of sounds like the use of reimannian concepts in relativity, it seems a little arbitrary and defines everything people are teaching. it sounds really big in QFT to describe everything as operators on hilbert spaces? later we'll be working with these operators at a higher level? i understand likely this was chosen because it is necessary and core, but i'm curious what QFT adds by this specific generalization of relations. what additional predictions does it make? what beginning states does one first learn about in qft? i'm curious what kind of states let one add or remove a particle -- are those infinite? Yes—it is a big generalization, but not just decorative abstraction. QFT keeps quantum superposition **and** makes spacetime-local fields fundamental, which lets particle number change while remaining compatible with special relativity. That is needed for processes such as particle production/decay and is the framework of particle physics.[^24_1][^24_2] The Hilbert-space/operator language does not itself predict a specific particle. The **specific field equation and interaction rules** determine predictions: masses, scattering probabilities, decay rates, etc. The first states to learn are: - **Vacuum**: no particles, written $|0\rangle$ - **One-particle state**: one excitation in a chosen mode, $a^\dagger|0\rangle$ - **Two-particle state**: $(a^\dagger)^2|0\rangle$ - And so on: $0,1,2,\ldots$ particles. Creation operators move up that list; annihilation operators move down.[^24_2] So yes: the full state space is infinite, because it allows arbitrarily many particles and, in continuum space, arbitrarily many possible modes. But each beginner example can be a tiny finite fragment: vacuum + one selected mode, then states labeled just by $0,1,2,\ldots$. <span style="display:none">[^24_10][^24_11][^24_12][^24_13][^24_14][^24_15][^24_3][^24_4][^24_5][^24_6][^24_7][^24_8][^24_9]</span> <div align="center">⁂</div> [^24_1]: https://plato.stanford.edu/archives/fall2018/entries/quantum-field-theory/in... [^24_2]: http://web1.eng.famu.fsu.edu/~dommelen/quantum/style_a/qft.html [^24_3]: https://pmc.ncbi.nlm.nih.gov/articles/PMC8623095/ [^24_4]: https://link.aps.org/doi/10.1103/PhysRev.132.2353 [^24_5]: https://mathoverflow.net/questions/211082/quantum-fields-and-infinite-tensor... [^24_6]: https://www.reddit.com/r/Physics/comments/198xk1p/precise_difference_between... [^24_7]: https://www.physicsforums.com/threads/where-exactly-does-qft-differ-from-qm-... [^24_8]: https://www.reddit.com/r/AskPhysics/comments/c13ltw/what_experiment_gave_bir... [^24_9]: https://physics.stackexchange.com/questions/209784/hilbert-space-of-a-quantu... [^24_10]: https://www.azoquantum.com/Article.aspx?ArticleID=545 [^24_11]: https://mathoverflow.net/questions/441769/meaning-of-a-quantum-field-given-b... [^24_12]: https://en.wikipedia.org/wiki/Hilbert_space [^24_13]: https://nicf.net/articles/qft-free-fields/ [^24_14]: https://www.quandela.com/resources/quantum-computing-glossary/hilbert-space/ [^24_15]: https://www.reddit.com/r/AskPhysics/comments/12w8nz6/why_do_we_use_hilbert_s... --- # hmm when interpreting your reply it's not clear to me where you are referring to different positions within a hilbert space, vs different basis states. what is the name of the state space you describe? what are its bases and dimensionality? The state space I described is called **Fock space**. For one chosen mode, its basis is: $$ |0\rangle,\ |1\rangle,\ |2\rangle,\ |3\rangle,\ldots $$ Each basis state means “exactly $n$ particles in this mode.” They are separate coordinate directions. A general state—one **position** in this space—is a combination: $$ c_0|0\rangle+c_1|1\rangle+c_2|2\rangle+\cdots $$ Its dimension is infinite because there are infinitely many basis states, one for each possible particle count.[^25_1] Creation maps $|n\rangle$ toward $|n+1\rangle$; annihilation maps it toward $|n-1\rangle$. Real QFT also has infinitely many modes, making the full Fock space larger still. <span style="display:none">[^25_2][^25_3][^25_4][^25_5][^25_6][^25_7][^25_8][^25_9]</span> <div align="center">⁂</div> [^25_1]: https://en.wikipedia.org/wiki/Hilbert_space [^25_2]: https://link.aps.org/doi/10.1103/PhysRevD.111.045001 [^25_3]: https://www.authorea.com/doi/pdf/10.22541/au.175449372.28574879 [^25_4]: https://www.reddit.com/r/QuantumPhysics/comments/q3f1ta/what_is_contained_in... [^25_5]: https://www.youtube.com/watch?v=GYv46_PTuw0 [^25_6]: https://physics.stackexchange.com/questions/65473/quantum-field-theory-and-h... [^25_7]: https://www.facebook.com/groups/1685507044817357/posts/28247412308200136/ [^25_8]: https://arxiv.org/html/2308.06669v3 [^25_9]: https://ncatlab.org/nlab/show/Hilbert+space --- # - in learning a Fock space describes one "mode" and a position lists a degree to which any particular count of particles are in that mode, similar (or identical?) to a statistical distribution - in QFT a fock space describes infinitely many modes and thus has a higher kind of infinity to its dimensionality, a 2-dimensional infinite plane of basis states? First bullet: **almost.** For one mode, the coordinates $c_0,c_1,\ldots$ are *amplitudes*, not probabilities. Their squared magnitudes $|c_n|^2$ give the probability of measuring $n$ particles, after normalization. So it resembles a probability distribution, but contains extra phase information. Second: don’t picture a 2-D infinite plane. For many modes, a basis state is labeled by a whole **occupancy list**: $$ |n_1,n_2,n_3,\ldots\rangle $$ meaning $n_i$ particles in mode $i$. The space remains simply **infinite-dimensional**; “higher kind of infinity” is not useful here. In the standard setup it can still have a countable orthonormal basis.[^26_1][^26_2] <span style="display:none">[^26_10][^26_11][^26_12][^26_13][^26_3][^26_4][^26_5][^26_6][^26_7][^26_8][^26_9]</span> <div align="center">⁂</div> [^26_1]: https://en.wikipedia.org/wiki/Fock_space [^26_2]: https://www.fuw.edu.pl/~kostecki/daniel_essay.pdf [^26_3]: https://link.aps.org/doi/10.1103/PhysRevD.76.125005 [^26_4]: https://arxiv.org/html/2506.17710v3 [^26_5]: https://www.sciencedirect.com/science/article/pii/S0550321325003281 [^26_6]: https://www.worldscientific.com/worldscibooks/10.1142/10367 [^26_7]: https://www.sciencedirect.com/topics/computer-science/fock-space [^26_8]: https://www.reddit.com/r/askmath/comments/3p9uuw/eli5_what_is_the_fock_space... [^26_9]: https://www.facebook.com/groups/695173879704511/posts/735930208962211/ [^26_10]: https://www.youtube.com/watch?v=zjuDDEp-pvw [^26_11]: https://mathoverflow.net/questions/302380/the-fock-space-vs-the-hilbert-spac... [^26_12]: https://physics.stackexchange.com/questions/617834/is-the-fock-space-defined... [^26_13]: https://mathoverflow.net/questions/211082/quantum-fields-and-infinite-tensor...