# What is a field, really?

There was a reason why I constantly failed physics at school and university, and that reason was, apart from the fact I was immensely lazy, that I mentally refused to "believe" more advanced stuff until I understand the fundamentals (which I, eventually, never did).

As such, one of the most fundamental things in physics that I still don't understand (a year after dropping out from the university) is the concept of field. No one cared to explain what a field actually is, they just used to throw in a bunch of formulas and everyone was content. The school textbook definition for a field (electromagnetic in this particular case, but they were similar), as I remember it, goes like:

An electromagnetic field is a special kind of substance by which charged moving particles or physical bodies with a magnetic moment interact.

A special kind of substance, are they for real? This sounds like the authors themselves didn't quite understand what a field is so they decided to throw in a bunch of buzzwords to make it sounds right. I'm fine with the second half but a special kind of substance really bugs me, so I'd like to focus on that.

### Is a field material?

Apparently, it isn't. It doesn't consist of particles like my laptop or even the light.

If it isn't material, is it real or is it just a concept that helps to explain our observations? While this is prone to speculations, I think we can agree that in scope of this discussion particles actually do exist and laws of physics don't (the latter are nothing but human ideas so I suspect Universe doesn't "know" a thing about them, at least if we're talking raw matter and don't take it on metalevel where human knowledge, being a part of the Universe, makes the Universe contain laws of physics). Any laws are only a product of human thinking while the stars are likely to exist without us homo sapiens messing around. Or am I wrong here too? I hope you already see why I hate physics.

### Is a field not material but still real?

Can something "not touchable" by definition be considered part of our Universe by physicians? I used to imagine that a "snapshot" of our Universe in time would contain information about each particle and its position, and this would've been enough to "deseralize" it but I guess my programmer metaphors are largely off the track. (Oh, and I know that the uncertainty principle makes such (de)serialization impossible — I only mean that I thought the Universe can be "defined" as the set of all material objects in it). Is such assumption false?

At this point, if fields indeed are not material but are part of the Universe, I don't really see how they are different from the whole Hindu pantheon except for perhaps a more geeky flavor.

When I talked about this with the teacher who helped me to prepare for the exams (which I did pass, by the way, it was before I dropped out), she said to me that, if I wanted hardcore definitions,

a field is a function that returns a value for a point in space.

Now this finally makes a hell lot of sense to me but I still don't understand how mathematical functions can be a part of the Universe and shape the reality.

I'm afraid the question will seem ambiguous and get closed but I had a really hard time trying to shape my confusion into sentences, and I will highly appreciate if someone clears it up, suggests a link or confirms that there is no definitive answer to my question.

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Not speaking as a moderator here, but both the wall of text and the heavy use of boldface make this look and feel like a rant before I even read a word. –  dmckee Aug 3 '11 at 23:52
Oh. Exactly because it is a wall of text I thought it's better to mark some parts in bold. I'll remove the boldface then. –  Dan Aug 4 '11 at 0:04
To the point how "real" a field is, you could take into account that for example the EM-field has momentum and energy,... See the Feynman lectures Vol 1. page 10-9 for a reference. –  student Sep 5 '12 at 20:28

I'm going to go with a programmer metaphor for you.

• The mathematics (including "A field is a function that returns a value for a point in space") are the interface: they define for you exactly what you can expect from this object.

• The "what is it, really, when you get right down to it" is the implementation. Formally you don't care how it is implemented.

In the case of fields they are not matter (and I consider "substance" an unfortunate word to use in a definition, even though I am hard pressed to offer a better one) but they are part of the universe and they are part of physics.

What they are is the aggregate effect of the exchange of virtual particles governed by a quantum field theory (in the case of E&M) or the effect of the curvature of space-time (in the case of gravity, and stay tuned to learn how this can be made to get along with quantum mechanics at the very small scale...).

Alas I can't define how these things work unless you simply accept that fields do what the interface says and then study hard for a few years.

Now, it is very easy to get hung up on this "Is it real or not" thing, and most people do for at least a while, but please just put it aside. When you peer really hard into the depth of the theory, it turns out that it is hard to say for sure that stuff is "stuff". It is tempting to suggest that having a non-zero value of mass defines "stuffness", but then how do you deal with the photo-electric effect (which makes a pretty good argument that light comes in packets that have enough "stuffness" to bounce electrons around)? All the properties that you associate with stuff are actually explainable in terms of electro-magnetic fields and mass (which in GR is described by a component of a tensor field!). And round and round we go.

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This is an amazing, thoughtful answer with a great metaphor, but no less than I expected on an SE site. So what you're saying is a “field” is like an interface, a contract, and the actual real world “implementation” is provided by the theory currently considered correct by the physicians for this type of field, of course, as long as we don't try to define “real world” which is kinda out of scope. Do I get it right? –  Dan Aug 4 '11 at 0:16
dmckee can come along to confirm or deny this, but if I'm understanding the metaphor correctly I think a physical theory would be akin to the API documentation. We are trying to write ourselves some documentation (i.e. develop the theory) by tinkering with the software (nature), without knowing anything about the implementation. (Of course this whole process is complicated by the fact that we exist in the software, but that's another matter entirely.) –  David Z Aug 4 '11 at 6:56
By the way, @dmckee, kudos on saving what could have been a sketchy question with a very good answer. –  David Z Aug 4 '11 at 6:56
@David I think that "writing the documentation" takes the metaphor further than I had, but that it is a very good description of what were doing as we do science. –  dmckee Aug 4 '11 at 13:04
@Newman, conventionally when people use the word "matter" they are talking about massive things, so massless entities like the electromagnetic field are excluded. –  Doug Packard Sep 6 '12 at 6:49

You say:

she said to me that, if I wanted hardcore definitions,

a field is a function that returns a value for a point in space.

Now this finally makes a hell lot of sense to me but I still don't understand how mathematical functions can be a part of the Universe and shape the reality.

You don't have to use super-complicated examples such as electromagnetism. I'll give you two examples which I hope will make it more clear; let me know if this helps.

## Example 1: Temperature

You might have come across that the higher you climb (on Earth or somewhere else, but let's think of Earth) the colder the air gets, at an typical rate of about 6ºC per kilometer (it depends on various factors, but this is a ballpark value); in meteorology, this is known as the lapse rate: the rate of temperature drop with altitude.

Now suppose you're observing a large, uniform terrain (e.g. a "flat desert"). If you want to ask:

What is the temperature of the air at a point $(x,y,z)$?

then you'll ascribe a certain value of temperature for each point. But to make a "table" to give the temperature for every point is certainly impractical! You try instead to use a function, an application, that gives the value of the temperature for each point: $$f : (x,y,z) \mapsto f(x,y,z)$$ I'll use a clearer nomenclature: $$T : (x,y,z) \mapsto T(x,y,z)$$ So this is a function with arguments in a $\mathcal{R}^3$ space (three-dimensional space, $\mathcal{R}\times\mathcal{R}\times\mathcal{R}$) which gives values in a 1-dimensional $\mathcal{R}$ space. Those values represent the values of the temperature at each coordinate $(x,y,z)$ of $\mathcal{R}^3$. Instead of writing $T(x,y,z)$ you can be more "practical" and write just $T$ as shorthand (especially when you're some calculus in an exercise).

That function represents a field -- the temperature field.

### "But what's the use of that?!"

What does it look like? If you have the ideal case of a perfectly flat "desert" and an idealized atmosphere, the temperature field will be something like: $$T(x,y,z) = T(x,y,z_0) - \frac{dT}{dz} (z-z_0)$$ Some notes:

1. In this situation, the temperature only varies in the vertical; it looks the same at any place over the desert -- there is really no depedence in the coordinates $x$ and $y$. Because of that you could make it easier for you and shorten the expression to just $T(z) = T(z-z_0) - dT/dz$.
2. In case you don't know/forgot: $dT$ is how much the temperature $T$ varies when you increase your height by a small (infinitesimal!) amount $dz$.
3. Don't worry about the minus sign next to the rate. It's put there by hand to have the expected physical meaning. When you go from a height level $z$ to $z+dz$, the temperature should decrease, from $T$ to $T-dT$ where $-dT < 0$, so that $-dT/dz$ is negative (it "takes away" from the temperature as you increase the altitude $z$). Example: from $z=1000$ to $z+dz = 1001$, the temperature should drop from $T$ to $T-0.006$ where $T$ is the temperature at level $z=1000$. Of course, that small value is because $0.006/(1001-1000) = dT/(dz+z-z) = dT/dz = 6$ Celsius per km.
4. I've intentionally abused the expression above to make it easier to understand. A more appropriate expression would be (if you've studied "integrals" in calculus) something like $$T(x,y,z) = T(x,y,z_0) - \int\limits_{z_0}^z \frac{dT}{dz} dz\ .$$

You have to give the temperature at a certain level $z_0$ of your choice to represent a specific case; it can be at the surface, $z_0 = 0\ \mathrm{meters}$. That function you have there represents the temperature field for that situation. If you have a "hot spot" -- e.g. you light up a candle -- then the temperature distribution (the field!) will be different, and the mathematical expression to describe the temperature field will be different (more complicated).

So this temperature field describes what is the temperature over that "desert air". It represents a quantity which has a spatial distribution. You can make it much more shorthanded if you just ignore the frontier condition $T(z_0)$ at a certain vertical level $z_0$ (which is arbitrary!) and write the field as $$-\frac{dT}{dz}\ .$$

## Example 2: Wind velocity

The example above illustrates a scalar field: the value of the field at each space point takes a scalar value ("just a number"). Not all fields are scalar. An example is the velocity field, which represents the velocity (direction and magnitude!) of the air at each point.

You can write it as $$\vec v : (x,y,z) \mapsto \vec v(x,y,z)$$ and for each point $(x,y,z)$ it describes what is the direction and magnitude of the air displacement at that point, the vector $\vec v$ at that point.

### What does it look like?

(The mathematical expression?) Well, it will depend on the situation of course! The expression can be impossibly complicated to write analytically. You certainly won't write the velocity field (or the temperature field) for the air inside your living room -- it's too complicated to write a mathematical expression! The best you can do is

1. Know a few laws or expressions or (more correctly) models, perhaps deduced from first principles, to describe how the conditions of a tiny piece of air will be influenced by the conditions of the neighbouring regions. Those models can be very simple or more elaborate; in the latter for meteorology, you just use computers to do the complicated ballance for each and every "air cell". In the example 1 with the temperature above, there is no horizontal dependence, but the rate at which the temperature varies vertically depends on the temperature, pressure, etc on top of the "tiny air box/cell/element" and on the bottom -- those are the ones who produce an effect.

2. Make some simplifications about the initial conditions, such as knowing what is the temperature along the walls and assuming (for example) there aren't "hot spots" or if there are, they're too insignificant to spot the difference against the situation where there aren't hot spots.

## Example 3: the electromagnetic field

When you put an electrically-charged tiny particle (test particle) near a metalic place (for example) that has an electric charge itself (like the plate of a large capacitator, for example), in the most general and broad case the force that the particle will feel will depend on where the particle is relative to the charged plate.

The force the test particle feels has a magnitude as well as a direction. If you put the test particle in another position, if will feel the force with a different intensity and direction.

You could put the test particle in many different places around the plate and measure the electric force felt by the test particle. And you collect the direction and intensity of that force. If you are able to condense that description of the magnitudes and directions of the electric force felt by the particle, you're writing it as a field, $$\vec E : (x,y,z) \mapsto \vec E(x,y,z)\ .$$

You can interpret the electromagnetic field as nothing more as a "mash-up" of both the electric force and magnetic force that a test particle will feel at each point of space.

## OK, but can you "touch" a field?

As a final note, I'll say the following. This question is more subject to discussion. Personally, I don't quite think about "touching" a field or it being "material"; I don't know how you're supposed to "touch" temperature.

The field represents the set of values for a quantity on a given space, and thus we arrive at your teacher's comment. In the classical physics sense that I've presented above, you can interpret the fields as "our way" of describing something that it's there, in a shorthand (a mathematical expression instead of a "spreadsheet of values"). In that case, I see the concept of field mixing up with the "thing" that it's representing. I won't debate that because I'm not sure I can explain better.

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From the way fields are actually used in physics and engineering, and consistent with the mathematical definition, fields are properties of any extended part of the universe with well-defined spatial boundaries. (The latter may be missing in case of infinitely extended objects, e.g., the universe as a whole - if it is infinitely extended.)

Causality is reflected in the fact (that makes physical predictions - and indeed life, which is based on the predictability of Nature - possible) that to a meaningful (and sometimes extremely high) accuracy, changes with time in the complete set of fields relevant for a particular application are determined by the current values of these fields.

Being properties of objects, fields cannot be touched but they can be sensed by appropriate sensors. In particular, several human senses probe properties of fields close the surface of the corresponding sensors:

• Eyes for sensing oscillations of the electromagnetic field passing through the lense,
• ears for (a) sensing oscillations of the pressure field of the air and (b) sensing the direction of the gravitational field,
• the skin for sensing stress fields and temperature fields close to the body surface,
• the tongue for sensing chemical concentration fields close to the surface of the tongue.

More specifically, a field is a numerical property of an extended part of the universe, which depends on points characterized by position and time (though the time dependence may be trivial). It is called a scalar, vector, tensor, operator field etc., depending on whether the numerical values at each point are scalars, vectors, tensors, operators, etc., and a real or complex field depending on whether these objects have real or complex coefficients.

Fields are the natural means to characterize numerically the detailed properties of extended macroscopic objects. This can be seen on a very elementary level. (It also applies to microscopic objects, but there the characterization is much more technical.)

All macroscopic objects possess a number of fields, most of them natural in the sense that all humans in our current technological culture experience in their daily life aspects of these fields either with their own sensors, or with technical gadgets known to be sensitive to these.

• always a scalar mass density field telling how the mass of the object is distributed in space and changes with time,
• in case of uneven composition such as rocks, concentration fields of the various chemical substances it contains.
• in case of nonrigid objects such as fluids, a vector velocity field (or several for each chemical substance), describing the local velocity of the mass flow.
• always a scalar temperature field telling how the temperature of the object is distributed in space and changes with time,
• always a stress tensor field telling how the mechanical forces inside the object are distributed in space and changes with time.

• in case of electrically active objects such as coils or capacitors, a scalar charge density field telling how the charge of the object is distributed in space and changes with time, and a vector current field describing the local velocity of the charge flow.

Not tangible objects such as the space between material objects also have space-time dependent properties, and hence associated fields, namely the (in nonrelativistic case scalar) gravitational field, the (vector) electric field and the (vector) magnetic field.

Hardly visible in everyday life, but very important in physics is an additional field, the (scalar) energy density field telling how the internal energy of the object is distributed in space and changes with time.

Additional fields are employed by physicists whenever the above fields are either not sufficient to give a complete description of the phenomenology they are interested in, or not sufficient to give a tractable theoretical description of the processes.

Causality is implemented by means of parabolic or hyperbolic differential equations relating the derivatives of the fields.

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I thought the concept of field was introduced in physics to make sure that the interaction is local, is this the same as to implement causality? –  Revo Mar 3 '12 at 19:46
- Fields were used long before such foundational concerns; already Euler used fields for his fluid equations. - Local interaction means precisely that dynamics is implemented by means of partial differential equations. - Causality means that information cannot flow backward in time and is an independent property: PDEs need not be causal, while many integro-differential equations are causal. –  Arnold Neumaier Mar 3 '12 at 20:07
I see. But can such non causal PDEs describe physical situations? –  Revo Mar 3 '12 at 20:58
Of course not! That's why I had written in my answer ''parabolic or hyperbolic'', which typically gives a causal dynamics, though one needs additional technical conditions to prove it. –  Arnold Neumaier Mar 3 '12 at 21:04

I think the history of the field concept helps to understand what it means today. There's a reference to Nancy J. Nersessian's paper ""Faraday's Field Concept" where she says:

The specific features of Faraday's field concept, in its 'favourite' and most complete form, are that force is a substance, that it is the only substance and that all forces are interconvertible through various motions of the lines of force. These features of Faraday's 'favourite notion' were not carried on. Maxwell, in his approach to the problem of finding a mathematical representation for the continuous transmission of electric and magnetic forces, considered these to be states of stress and strain in a mechanical aether. This was part of the quite different network of beliefs and problems with which Maxwell was working

So for Faraday and Maxwell, the field was seen as a substance responsible for transmitting the effects of force and so doing away with action at a distance.

Today, the field concept is still used to account for action at a distance as measurable effects which propagate between separate points, but without force or aether as a substance. Gauge field theories, for example, use particles called gauge bosons to transmit the fundamental forces of nature between separated particles.

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A heretic could wonder whether taking these virtual particles too seriously is as bad as maxwell's thinking about the aether.... –  joseph f. johnson Jan 15 '12 at 3:26

Field is tool which we use to implement the fact that Quantum Field Theory gives meaning to local measurements (those that can be performed in one space-time point, or more realistic, in small space-time region around that point). That is why fields depend on space-time coordinates. In canonical quantization, fields are self-adjoint operators, which means that these are observables in principle, and that all observables can be built from them. In Feynman path integral approach to quantization (which is equivalent to canonical), fields are ordinary functions of space-time coordinates, if they correspond to bosonic fields from canonical quantization. To fermionic operators from canonical formalism correspond Grasmann numbers in path integral approach. Fields are real in operational sense, that we can define them, build theory using them and make predictions for observations.

As far is known there is no Quantum Field Theory that works for any (arbitrarily) large energy scale, as it is possible that very short distance physics is best described by theory other than QFT (string theory, i.e). So asking about if field is real or not can only be put into point of view of some theory. Exact position and momentum of particle is real and useful concept in classical mechanics, for example, but in Quantum Mechanics this sort of determinism was lost.

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Ok there is only one good answer to this. It is found in a book Local Quantum Physics, by Rudolf Haag, master of QFT, page 46, first paragraph:

Yet the belief in field-particle duality as a general principle, the idea that to each particle there is a corresponding field and to each field a corresponding particle has also been misleading and served to veil essential aspects. The rôle of fields is to implement the principle of locality. The number and the nature of different basic fields needed in the theory is related to the charge structure, not to the empirical spectrum of particles. In the presently favoured gauge theories the basic fields are the carriers of charges called colour and flavour but are not directly associated to observed particles like protons.

More or less, it meets answer by Newman.

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Some remarks from the history of the field concept.

Kepler has invented his 3 laws explaining how planets orbit around the Sun.

Newton discovered his gravity law, that replaced Kepler's 3 laws. According to Newton, massive bodies attract each other at a distance.

Pierre-Simon de Laplace did not accept the concept of the "action at a distance". He assumed that:

1. Massive body generates a "substance" in space around it, and "intensity" of this "substance" depends on a distance to the massive body. Nowdays this intensity is called gravitational potential.

2. This "substance" acts on other massive body, and the acting force is proportional to the gradient of the intensity.

In other words, two massive bodies do not act directly on each other, but each body is a source of "substabce" called "field", and then field acts on other body.

What's the difference? It is as follows:

1. From the Newton's law Laplace derived equation for the field only (known as Laplace equation). The same field can be derived by various combinations of sources, and once you know the field (or, to be more precise, it's gradients) in the area where massive body is located, you don't need to care about the sources of this field in order to identify the force acting on the massive body at this area.

2. The concept of the "action at a distance" is not comptible with special relativity in a sense, that if the position of the massive body will change, it will not cause immediate change of the force acting on other body. Hence, with the concept of field modern physics describes this as following: change of the position of the massive body causes gradual chenge of the field, with the subsequent (but not immediate!) change of the force acting on other bodies.

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I enjoyed @dmckee answer but would like to add that fields are primarily a mathematical tool, with definite rules taken over by physicists in their effort to describe the observed world.

In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication.

I will also add that in classical physics, fields are defined as well, with no need of virtual particle exchanges to be useful in calculations .

And my third observation is that there exists the old pythgorean view of the world, that "God continually geometrises" ( if I may coin a word), or the so called "platonic ideals", in modern words : given the mathematics reality will follow the mold.

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You seem to have picked the wrong mathematical meaning of "field". That one refers to things like real or complex numbers, and has basically nothing to do with the issue at hand. Maybe you meant something like en.wikipedia.org/wiki/Scalar_field or en.wikipedia.org/wiki/Scalar_field#Other_kinds_of_fields –  Mark Eichenlaub Aug 9 '11 at 1:10
@Mark no, I mean what I linked: that there are mathematical definitions of the word "field" that physics took over. These definitions, like closure and commutativity, are a necessary underlying level to the "fields" of physics. Your links are the physical meaning, not the mathematical. –  anna v Aug 9 '11 at 4:39
Most of the physicist's "fields" will be defined over the algebraist's fields, but that's as deep as any relevance goes. It's not a corruption of the algebraic concept, and it's not even a purely physicists' term, since it's not like "vector field" would be unknown to anyone who takes elementary differential equations or even just basic calculus yet never steps foot in a physics classroom. To satisfy the mathematician's need for abstraction and generality, one one can say a (non-algebraic) "field" is a section of a fiber bundle... but I'm still at a loss as to where your point is going. –  Stan Liou Aug 9 '11 at 7:09
My point is simply that the definition of a field is more general than the domain of physics, and in physics also includes classical fields, omitted by the chosen answer. –  anna v Aug 9 '11 at 10:33
If you want to point out that field is more general than the domain of physics I think you can see it for example mathematically as a section of a fiber bundle as Stan Liou suggested. But I think the algebraic concept of a field (a commutative ring ...) you referred to has nothing to to with the physical field. So if you see any nontrivial relationship to the algebraic field please give some details. –  student Sep 5 '12 at 19:13

A body cannot directly exert force or interact with other bodies.

So there should be a connection between the two. Or a manipulation of the space itself.

So the manipulation of space can analytically be a field

I feel this is an open ended problem

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Ah, the idea of the field as interface. That's the whole philosophy of the source field method introduced by Schwinger. Treat the field as a perturbative source, and watch how it kicks the system under study. But always, at the end of the day, we have to take the limit as the source field goes to zero. The interface is fictitious after all, right? We only imagine the interface, or not?

And then there's the S-matrix approach introduced by Heisenberg for philosophical reasons, with the philosophy carried to extremes by Chew with his bootstrap model. Bypass fields entirely, and only focus upon the asymptotic past and asymptotic future. Only the asymptotic future is observable. But it's not like we're living in the asymptotic future, are we?

The interface analogy might work for a classical simulation, but not a quantum simulation. Quantum mechanics changes everything. Sure, for a classical simulation, the programmer can implement a read-only query function call which otherwise doesn't affect the simulation. But that's impossible for a quantum simulation, and our universe is quantum.

Say there's a programmer out there in heaven implementing the program simulating our universe. This programmer is the demiurge, a god but not God. Say this demiurge wishes to implement a read query for transcendent angels in heaven to call to query internal states of our universe. Now, let's say a transcendent angel Eve comes along and wants to eavesdrop our thoughts. What a snoop! Quantum information theory tells us Eve can't eavesdrop without affecting the system, i.e. our universe, and in principle, the mere act of calling the query can be detected within our universe. See, we would become entangled with Eve.

If both heaven and earth are quantum, epiphenominalism is impossible.

Or maybe Eve wishes to avoid detection by only accessing the query at the end of the simulation? Would that work? No, because of the retrocausal effects of the query. See delayed choice experiment. Take note, metaphysics-hating positivists, in principle, transcendent angels in heaven can produce measurable effects in our universe. Or if Eve is really careful, she can avoid snooping whenever there are experimenters trying to measure her effects? Isn't that a common complaint heard by skeptics?

OK, maybe the demiurge wanted to be deistic and not implement any read query interfaces. But that demiurge would also have to prevent retrocausal influences from the end of the simulation. That can only be done if the end result of the quantum simulation is exactly uncomputed, erasing everything. From nothing, back to nothing. We might as well not even exist, and no one in heaven will know about us.

So maybe the demiurge wasn't deistically inclined and left trapdoors to allow angels to influence our world with their will.

Or maybe the demiurge simulated our quantum universe on a classical computer. Let's just assume he has exponential resources to waste. But to read our universe, he still has to choose a decoherent histories realm anyway, anthropically selected upon our existence and conditioned upon other factors as well, like our history. Heaven is still detectable.

The lesson of quantum mechanics is there has to be a transcendent observer observing us.

A closed quantum system of which we are a part can never have definite outcomes, and more importantly, doesn’t have and can't have a preferred basis. A quantum universe can never be closed if it were real.

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You may have take the wrong meaning of "interface" from my answer. That was explicitly a metaphor for programmers who may be assumed to be familiar with the interface--implementation distinction used to reduce the number of interdependency of modules (and pushed heavily in OOP, though it has been floating around for a long time). –  dmckee Aug 9 '11 at 3:08

## protected by Qmechanic♦Jul 6 '13 at 12:39

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